Quartz crystal resonator, quartz crystal unit, and quartz crystal oscillator

ABSTRACT

A quartz crystal unit comprising a quartz crystal resonator having a base portion, and first and second tuning fork arms connected to the base portion, the base portion having a length less than 0.5 mm and greater than a spaced-apart distance between the first and second tuning fork arms, each of the first and second tuning fork arms having a width less than 0.1 mm and a length less than 1.56 mm, and a plurality of different widths including a first width and a second width greater than the first width, at least one groove being formed in at least one of opposite main surfaces of each of the first and second tuning fork arms so that a length of the at least one groove is within a range of 0.3 mm to 0.79 mm, the quartz crystal resonator being housed in a case, and a lid being connected to the case.

FIELD OF THE INVENTION

The present invention relates to a resonator, a unit having theresonator, an oscillator having the unit and an electronic apparatushaving the oscillator.

BACKGROUND OF THE INVENTION

There are many electronic apparatuses comprising a display portion and aquartz crystal oscillator at least. For example, cellular phones,wristwatches, facsimiles, digital cameras and DVD recorders comprising aquartz crystal oscillator are well known. Recently, because of highstability for frequency, miniaturization and the light weight nature ofthese electronic apparatuses, the need for an electronic apparatuscomprising a smaller quartz crystal oscillator with a frequency of highstability has arisen. For example, the quartz crystal oscillator havinga quartz crystal tuning fork resonator housed in a unit, which vibratesin a flexural mode, is widely used as a time standard in an electronicapparatus such as the cellular phones, the wristwatches, the facsimiles,the digital cameras and the DVD recorders.

Similar to this, the same need has also arisen for an electronicapparatus comprising a contour mode resonator such as alength-extensional mode quartz crystal resonator, a width-extensionalmode quartz crystal resonator and a Lame mode quartz crystal resonatoror a thickness shear mode quartz crystal resonator or a SAW (SurfaceAcoustic Wave) resonator or a resonator for sensing angular velocitymade of a piezoelectric material such as quartz crystal, lithiumtantalite (LiTaO₃), lithium niobate (LiNbO₃) and ceramics.

Heretofore, however, it has been impossible to obtain an electronicapparatus comprising a smaller quartz crystal oscillator with aminiature quartz crystal tuning fork resonator of the prior art, capableof vibrating in a flexural mode, and having a frequency of highstability, a small series resistance and a high quality factor. This isthe reason why, when miniaturized, the quartz crystal tuning forkresonator of the prior art, capable of vibrating in a flexural mode hasa smaller electromechanical transformation efficiency. As a result, theresonator has a frequency of low stability, a large series resistanceand a reduced quality factor.

Additionally, there has been a big problem in the quartz crystaloscillator of the prior art having the quartz crystal tuning forkresonator of the prior art, such that a frequency of a fundamental modeof vibration of the tuning fork resonator which is an output signal ofthe oscillator jumps to a second overtone mode of vibration thereof byshock or vibration.

Similarly, however, it has been impossible to obtain an electronicapparatus comprising a smaller quartz crystal oscillator with a contourmode resonator such as a length-extensional mode quartz crystalresonator, a width-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator or a thickness shear mode quartz crystalresonator or a SAW resonator or a resonator for sensing angular velocityhaving a frequency of high stability, a small series resistance and ahigh quality factor because, when miniaturized, each resonator has asmall electromechanical transformation efficiency, as a result, afrequency of low stability, a large series resistance and a low qualityfactor, and also is not strong against shock.

It is, therefore, a general object of the present invention to provideembodiments of a quartz crystal resonator, a quartz crystal unit, aquartz crystal oscillator and an electronic apparatus of the presentinvention, which overcome or at least mitigate one or more of the aboveproblems.

SUMMARY OF THE INVENTION

The present invention relates to a resonator, a unit, an oscillator andan electronic apparatus comprising a display portion and a plurality ofoscillators, one of which comprises a quartz crystal oscillatorcomprising a quartz crystal oscillating circuit having an amplificationcircuit and a feedback circuit, and in particular, relates to a quartzcrystal resonator capable of vibrating in a flexural mode, a quartzcrystal unit having the quartz crystal resonator and a quartz crystaloscillator having the quartz crystal unit and having an output signal ofa frequency of high stability for a fundamental mode of vibration of thequartz crystal resonator, and also to a quartz crystal oscillator havinga suppressed second overtone mode of vibration of the quartz crystalresonator, in addition, relates to a quartz crystal oscillatorcomprising an another contour mode resonator such as alength-extensional mode resonator, a width-extensional mode resonatorand a Lame mode resonator or a thickness shear mode resonator, each madeof quartz crystal or a SAW resonator or a piezoelectric resonator forsensing angular velocity. The quartz crystal oscillator is, therefore,available for the electronic apparatus requiring a miniature quartzcrystal oscillator with high time accuracy and shock proof.

It is an object of the present invention to provide a miniature quartzcrystal resonator, capable of vibrating in a flexural mode, and having ahigh electromechanical transformation efficiency.

It is an another object of the present invention to provide a miniaturequartz crystal unit with a quartz crystal resonator, capable ofvibrating in a fundamental mode of vibration of a flexural mode, andhaving a high electromechanical transformation efficiency.

It is a further object of the present invention to provide a quartzcrystal oscillator with a miniature quartz crystal resonator, capable ofvibrating in a flexural mode, and having a frequency of high stability,a small series resistance R₁ and a high quality factor Q₁, whose nominalfrequency for a fundamental mode of vibration is within a range of 10kHz to 200 kHz. Especially, a frequency of about 32.768 kHz is veryavailable for a time standard of a frequency signal.

It is a still another object of the present invention to provide anelectronic apparatus comprising a display portion and a plurality ofoscillators.

According to one aspect of the present invention, there is provided aquartz crystal resonator comprising: a plurality of vibrational arms,each of the vibrational arms having a first main surface and a secondmain surface and side surfaces; and a base portion to which thevibrational arms are attached, in which the resonator has apiezoelectric constant e′₁₂ in the range of 0.1 C/m² to 0.19 C/m² in theabsolute value.

According to a second aspect of the present invention, there is provideda quartz crystal unit comprising: a quartz crystal resonator having abase portion and a plurality of vibrational arms attached to the baseportion; a case for housing the quartz crystal resonator; and a lid forcovering an open end of the case, each of the vibrational arms having afirst main surface and a second main surface opposite the first mainsurface and side surfaces, in which the quartz crystal resonator has acutting angle in the range of ZYlwt(−20° to)+20°/(−25° to)+25°/(−18° to+18°) and a piezoelectric constant e′₁₂ of the resonator is within arange of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a third aspect of the present invention, there is provideda quartz crystal oscillator comprising: a quartz crystal oscillatingcircuit comprising; an amplification circuit comprising a CMOS inverterand a feedback resistor, and a feedback circuit comprising a quartzcrystal resonator capable of vibrating in a flexural mode, a pluralityof capacitors and a drain resistor, the quartz crystal resonator beinghoused in a package comprising a case for housing the quartz crystalresonator and a lid for covering an open end of the case, andcomprising: a plurality of vibrational arms, each of the vibrationalarms having a first main surface and a second main surface opposite thefirst main surface and side surfaces; and a base portion to which thevibrational arms are attached, in which the quartz crystal resonator hasa cutting angle in the range of ZYlwt (−20° to +20°)/(−25° to)+25°/(−18°to +18°) and a piezoelectric constant e′₁₂ of the resonator is within arange of 0.1 C/m² to 0.19 C/m² in the absolute value.

According to a fourth aspect of the present invention, there is providedan electronic apparatus comprising a display portion and a plurality ofoscillators, one of the oscillators being a quartz crystal oscillatorcomprising: a quartz crystal oscillating circuit comprising; anamplification circuit having a CMOS inverter and a feedback resistor,and a feedback circuit having a quartz crystal tuning fork resonatorcapable of vibrating in a flexural mode of an inverse phase, a pluralityof capacitors and a drain resistor, the quartz crystal tuning forkresonator comprising a tuning fork base and a plurality of tuning forkarms connected to the tuning fork base, each of the tuning fork armshaving a first main surface and a second main surface opposite the firstmain surface and side surfaces, the quartz crystal tuning fork resonatorbeing housed in a package comprising a case for housing the resonatorand a lid for covering an open end of the case, in which the quartzcrystal tuning fork resonator has a fundamental mode of vibration and asecond overtone mode of vibration and the amplification circuit of thequartz crystal oscillating circuit has negative resistances −RL₁ and−RL₂ for the fundamental mode of vibration and the second overtone modeof vibration of the quartz crystal tuning fork resonator, in which anabsolute value of the negative resistances is defined by |−RL₁| and|−RL₂| and a ratio of the |−RL₁| and R₁ is greater than that of the|−RL₂| and R₂, where R₁ and R₂ represent a series resistance of thefundamental mode of vibration and the second overtone mode of vibrationof the quartz crystal resonator, respectively, in which an output signalof the quartz crystal oscillating circuit has an oscillation frequencyof the fundamental mode of vibration of the quartz crystal tuning forkresonator and is a clock signal which is used to display time at adisplay portion of the electronic apparatus, and in which the quartzcrystal tuning fork resonator has a cutting angle in the range of ZYlwt(−20° to)+20°/(−25° to)+25°/(−18° to +18°) and a piezoelectric constante′₁₂ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in theabsolute value.

Preferably, the piezoelectric constant e′₁₂ is within a range of 0.12C/m² to 0.19 C/m² in the absolute value.

Preferably, mounting arms protruding from the base portion comprise two.

Preferably, each of the vibrational arms has a groove having a firststepped portion and a second stepped portion.

Preferably, each of the vibrational arms has a through groove.

Preferably, a merit value M₂ of a second overtone mode of vibration ofthe quartz crystal resonator having vibrational arms and a base portionis less than 30.

Preferably, the quartz crystal oscillator with the quartz crystalresonator is constructed so that a ratio of an amplification rate α₁ ofthe fundamental mode of vibration and an amplification rate α₂ of thesecond overtone mode of vibration of the amplification circuit isgreater than that of a feedback rate β₂ of the second overtone mode ofvibration and a feedback rate β₁ of the fundamental mode of vibration ofthe feedback circuit, and a product of the amplification rate α₁ and thefeedback rate β₁ of the fundamental mode of vibration is greater than 1.

The present invention will be more fully understood by referring to thefollowing detailed specification and clarms taken in connection with theappended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general view of a quartz crystal plate from which a quartzcrystal resonator of the present invention is formed;

FIG. 2 shows a plan view of a quartz crystal resonator of a firstembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 3 shows an A-A′ cross-sectional view of the vibrational arms of thequartz crystal resonator in FIG. 2 ;

FIG. 4 shows an A-A′ cross-sectional view of another embodiment of thevibrational arms of the quartz crystal resonator in FIG. 2 ;

FIG. 5 shows a plan view of a quartz crystal resonator of a secondembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 6 shows a plan view of a quartz crystal resonator of a thirdembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 7 shows a B-B′ cross-sectional view of the vibrational arms of theresonator in FIG. 6 ;

FIG. 8 shows a plan view of a quartz crystal resonator of a fourthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 9 shows a plan view of a quartz crystal resonator of a fifthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 10 shows a plan view of a quartz crystal resonator of a sixthembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 11 shows a plan view of a quartz crystal resonator of a seventhembodiment of the present invention, and comprising a quartz crystaltuning fork resonator capable of vibrating in a flexural mode;

FIG. 12 shows a plan view of a width-extensional mode quartz crystalresonator constructing an electronic apparatus of the present invention;

FIG. 13(a) and FIG. 13(b) show a plan view of a thickness shear modequartz crystal resonator constructing an electronic apparatus of thepresent invention and a F-F′ sectional view of the resonator;

FIG. 14 shows a plan view of a Lame mode quartz crystal resonatorconstructing an electronic apparatus of the present invention;

FIG. 15(a) and FIG. 15(b) show a plan view of a resonator for sensingangular velocity constructing an electronic apparatus of the presentinvention and a G-G′ sectional view of the resonator;

FIG. 16(a) and FIG. 16(b) show a plan view of a quartz crystal resonatorof an eighth embodiment of the present invention and comprising a quartzcrystal tuning fork resonator, and a J-J′ sectional view of theresonator;

FIG. 17 shows a plan view of a quartz crystal unit of a first embodimentof the present invention and omitting a lid;

FIG. 18 shows a plan view of a quartz crystal unit of a secondembodiment of the present invention and omitting a lid;

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a thirdembodiment of the present invention;

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of afirst embodiment of the present invention;

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillatingcircuit constructing a quartz crystal oscillator of the presentinvention;

FIG. 22 shows a diagram of the feedback circuit of FIG. 21 ; and

FIG. 23 shows a block diagram of an embodiment of an electronicapparatus of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring now to the drawings, the embodiments of the present inventionwill be described in more detail.

FIG. 1 is a general view of a quartz crystal plate 1 from which a quartzcrystal resonator of the present invention is formed, and particularly,a relationship of cutting angles θ_(x), θ_(y) and θ_(z) of the quartzcrystal plate 1 and its coordinate system is illustrated in FIG. 1 . Thecoordinate system has original point o, electrical axis x, mechanicalaxis y and optical axis z of quartz crystal and o-xyz is constructed.

First, a quartz crystal plate perpendicular to z axis, so called, Zplate quartz crystal is taken. The Z plate quartz crystal has adimension of Width W₀, length L₀ and thickness T₀ corresponding to arespective direction of x, y and z axes.

Next, this Z plate quartz crystal is, first, rotated with an angle θ_(y)about the y axis, second, rotated with an angle θ_(x) about x′ axiswhich is a new axis of the x axis, and third, rotated with an angleθ_(z) about z″ axis which is a new axis of the z axis. In this case,each of the x, y and z axes changes to x″, y″ and z″ axes, respectively,because each axis is rotated twice about two axes. A quartz crystalresonator of the present invention is, therefore, formed from the quartzcrystal plate with the rotation angles.

In other words, according to an expression of IEEE notation, a cuttingangle of the quartz crystal resonator of the present invention can beexpressed by ZYlwt(θ_(y))/(θ_(x))/(θ_(z)), and each of the angles θ_(y),θ_(x), θ_(z) will be described later in detail according to resonatorsof the present invention.

FIG. 2 shows a plan view of a quartz crystal resonator 10 of a firstembodiment of the present invention and which is a quartz crystal tuningfork resonator. The resonator 10 comprises vibrational arms 20 and 31and a base portion 40 attached to the vibrational arms, and the baseportion 40 has mounting arms 36 and 37 protruding from the base portion,each of which is mounted on a mounting portion of a package comprising acase for housing the resonator and a lid for covering an open end of thecase.

In addition, a length of the base portion 40 is greater than aspaced-apart distance between the vibrational arms 20 and 31 to get theresonator without a vibration leakage which is caused by vibration.

In more detail, the mounting arm 36 (first mounting arm) having a lengthL₃ and a width W₆ is connected to the base portion 40 having a widthW_(H) through a connecting portion 34 having a length L₂ and a widthW_(S) and the mounting arm 37 (second mounting arm) having a length L₃and a width W₆ is connected to the base portion 40 through a connectingportion 35 having a length L₂ and a Width W_(S). Namely, the connectingportion 34 is connected to the mounting arm 36 and the connectingportion 35 is connected to the mounting arm 37. In addition, as shown inFIG. 2 and FIG. 3 , an electrode 23 is disposed on a surface of a groove 21 of the vibrational arm 20 and an electrode 24 is disposed on asurface of a groove 22 of the vibrational arm 20, and electrodes 25, 26are disposed on side surfaces of the vibrational arm 20 so that theelectrodes 23, 24 have an electrical polarity opposite to an electricalpolarity of the electrodes 25, 26, similar to this, an electrode 29 isdisposed on a surface of a groove 27 of the vibrational arm 31, and anelectrode 30 is disposed on a surface of a groove 28 of the vibrationalarm 31, and electrodes 32, 33 are disposed on side surfaces of thevibrational arm 31 that the electrodes 29, 30 have an electricalpolarity opposite to an electrical polarity of the electrodes 32, 33,namely, the electrodes 23, 24 have the same electrical polarity as theelectrodes 32, 33, and the electrodes 25, 26 have the same electricalpolarity as the electrode 29, 30, and the electrodes 23, 24, 32, 33 havean electrical polarity opposite to an electrical polarity of theelectrodes 25, 26, 29, 30. In addition, as shown in FIG. 2 , theelectrode 23 disposed on the surface of the groove 21 of the vibrationalarm 20 is connected to an electrode disposed on a sur ace of themounting arm 36 through an electrode disposed on a surface of the baseportion 40 and an electrode disposed on a surface of the connectingportion 34, and ale electrode 29 disposed on the surface of the groove27 of the vibrational arm 31 is connected to an electrode disposed on asurface of the mounting arm 37 through an electrode disposed on asurface of the base portion 40 and an electrode disposed on a surface ofthe connecting portion 35. In other words, each of the vibrational arms20, 31 has a first main surface and a second main surface opposite thefirst main surface, a groove is formed in each of the first and secondmain surfaces of each of the vibrational arms 20, 31, and an electrodeis disposed on a surface of the groove formed in each of the first andsecond main surfaces of each of the vibrational arms 20, 31. From FIG. 2, the electrode disposed the surface of the groove formed in at leastone of the first and second main surfaces of one of the vibrational arms20, 31 is connected to the electrode disposed on the surface of themounting arm 36 through the electrode disposed on the surface of thebase portion 40 and the electrode disposed on the surface of theconnecting portion 34, while the electrode disposed on the surface ofthe groove formed in at least one of the first and second main surfacesof the other of the vibration arms 20, 31 is connected to the electrodedisposed on the surface of the mounting arm 37 through the electrodedisposed an the surface of the base portion 40 and the electrodedisposed on the surface of the connecting portion 35, In addition, asshown FIG. 17 , a case 165 has a mounting portion 166 and a mountingportion 167, and the electrode of the mounting arm 36 is connected to anelectrode disposed on a surface of the mounting portion 166 of the case165 and the electrode of the mounting arm 37 is connected to anelectrode disposed on a surface of the mounting portion 167 of the case165 so that the electrode of the mounting portion 166 has an electricalpolarity opposite to an electrical polarity of the electrode of themounting portion 167. In another expression, the electrode 23 of thegroove 21 of the vibrational arm 20 is connected to an electrodedisposed on a surface of one of the mounting arms 36, 37 through anelectrode disposed on a surface of one of the connecting portions 34, 35and an electrode disposed on a surface of the base portion 40, while theelectrode 29 of the groove 27 of the vibrational arm 31 is connected toan electrode disposed on a surface of the other of the mounting arms 36,37 through an electrode disposed on a surface of the other of theconnecting portions 34, 35 and an electrode disposed on a surface of thebase portion 40.

In addition to this, the width W_(S) of the connecting portion 34 isgreater than the width W₆ of the mounting arm 36 and the width W_(S) ofthe connecting portion 35 is greater than the width W₆ of the mountingarm 37.

Therefore, the base portion 40 and the connecting portions 34, 35 have athird mounting arm including a first width which is given as(W_(H)+2W_(S)) and the length L₂. Namely, the third mounting arm has thefirst width given as (W_(H)+2W_(S)) and the length L₂. In other words,an outer side surface of the third mounting arm has the first widthgiven as (W_(H)+2W_(S)). In addition, the third mounting arm comprisesan inner side surface having a width which is defined by the sum of thewidth W_(H) of the base portion and the width W_(S) of each of theconnecting portions 34, 35, the width of the inner side surface of thethird mounting arm has also the first width given as (W_(H)+2W_(S)).

In the first embodiment, as shown in FIG. 2 , the mounting arm 36 is notconnected to the mounting arm 37. However, this invention is not limitedto this, but includes the resonator 10 comprising a fourth mounting armthrough which the mounting arm 36 is connected to the mounting arm 37.In detail, the fourth mounting arm has a length substantially equal tothe length L₂ of the third mounting arm and a second width substantiallyequal to the first width given as (W_(H)+2W_(S)) so that the mountingarms 36, 37 have a frame with the third mounting arm including the baseportion 40, the connecting portions 34, 35 and the fourth mounting arm,namely, the frame comprises the mounting arms 36, 37, the third mountingarm and the fourth mounting arm. In detail, the vibrational arms 20, 31are surrounded by the frame comprising the mounting arms 36, 37, thethird mounting arm and the fourth mounting arm so that the vibrationalarms 20, 31 are located in the frame.

In addition, the fourth mounting arm is actually formed substantiallyparallel to the third mounting arm so that a direction of the secondwidth of the fourth mounting arm corresponds to a direction of a widthof each of the vibrational arms 20, 31. However, this invention is notlimited to this, but includes the third and fourth mounting arms thatthe fourth mounting arm is formed not parallel to the third mountingarm, namely, a direction of the first width of the third mounting arm isdifferent from that of the second width of the fourth mounting arm.

In this embodiment, though the length of the fourth mounting arm issubstantially equal to the length L₂ of the third mounting arm, thelength of the fourth mounting arm may be different from the length ofthe third mounting arm so that the length of the fourth mounting arm isgreater or less than the length L₂ of the third mounting arm. Similar tothis, though the second width of the fourth mounting arm issubstantially equal to the first width of the third mounting arm, thesecond width of the fourth mounting arm may be different from the firstwidth of the third mounting arm so that the second width of the fourthmounting arm is greater or less than the first width of the thirdmounting arm. In actual, a ratio of the length of the fourth mountingarm and the length L₂ of the third mounting arm is within a range of0.45 to 1.8, preferably, greater than 0.45 and less than 1 or greaterthan 1 and less than 1.8.

In order to get the resonator with a small series resistance R₁, thevibrational arms 20, 31 are housed in a vacuum, therefore, each of themounting arms 36, 37, and the third and fourth mounting arms has a firstmain surface (upper surface) and a second main surface (lower surface)opposite the first main surface and is connected to a case and a lideach of which has a concave portion including a depth within a range of0.01 mm to 0.2 mm, preferably, within a range of 0.015 mm to 0.18 mm,more preferably, within a range of 0.02 mm to 0.15 mm and four mountingportions including first, second, third and fourth mounting portions toget a unit comprising the resonator, the case and the lid. Also, each ofthe case and the lid is made of such a crystal plate as a quartz crystalplate and the concave portion of each of the case and the lid is formedin such an etching process as a chemical etching process, a mechanicaletching process and a physical etching process. In addition, the depthof the case is substantially equal to the depth of the lid, and each ofthe case and the lid has the depth greater than one-sixth and less thanthree-fourth of a thickness of the corresponding one of the case and thelid, preferably, within a range of one-fifth to one-half of thethickness. However, the depth of the case may be different from thedepth of the lid so that the depth of the case is greater or less thanthe depth of the lid.

Namely, the mounting arm 36 is mounted on the first mounting portion ofeach of the case and the lid so that the mounting arm 36 is locatedbetween the first mounting portion of the case and the first mountingportion of the lid. In other words, the first main surface of themounting arm 36 is connected to the first mounting portion of the caseand the second main surface of the mounting arm 36 is connected to thefirst mounting portion of the lid.

Similar to this, the mounting arm 37 is mounted on the second mountingportion of each of the case and the lid so that the mounting arm 37 islocated between the second mounting portion of the case and the secondmounting portion of the lid. In other words, the first main surface ofthe mounting arm 37 is connected to the second mounting portion of thecase and the second main surface of the mounting arm 37 is connected tothe second mounting portion of the lid.

In addition, the third mounting arm is mounted on the third mountingportion of each of the case and the lid so that the third mounting armis located between the third mounting portion of the case and the thirdmounting portion of the lid. In other words, the first main surface ofthe third mounting arm is connected to the third mounting portion of thecase and the second main surface of the third mounting arm is connectedto the third mounting portion of the lid.

Similar to this, the fourth mounting arm is mounted on the fourthmounting portion of each of the case and the lid so that the fourthmounting arm is located between the fourth mounting portion of the caseand the fourth mounting portion of the lid. In other words, the firstmain surface of the fourth mounting arm is connected to the fourthmounting portion of the case and the second main surface of the fourthmounting arm is connected to the fourth mounting portion of the lid.

Thus, each of the mounting arms 36, 37, and the third and fourthmounting arms is mounted on the corresponding one of the first, second,third and fourth mounting portions of each of the case and the lid. Itis, therefore, needless to say that a length dimension of each of themounting arms 36, 37 is substantially equal to a length dimension ofeach of the case and the lid, and also a width dimension of each of thethird and fourth mounting arms is substantially equal to a widthdimension of each of the case and the lid.

In addition, each of the mounting arms 36, 37 has outer side surfacesincluding the length L₃ and the width W₆, also, the fourth mounting armhas an outer side surface and an inner side surface opposite the outerside surface, and each of the inner and outer side surfaces of thefourth mounting arm has the second width substantially equal to thefirst width (W_(H)+2W_(S)), therefore, when each of the mounting arms36, 37 has the outer side surfaces including the length L₃ and the widthW₆, the third mounting arm has the outer side surface including thefirst width (W_(H)+2W_(S)) and the fourth mounting arm has the outerside surface including the second width substantially equal to the firstwidth (W_(H)+2W_(S)), an overall length dimension of the outer sidesurface of each of the mounting arms 36, 37, and the third and fourthmounting arms have a first overall length dimension which is given as2(L₃+2W₆+W_(H)+2W_(S)).

Similar to this, each of the mounting arms 36, 37 has an inner sidesurface including a length of L₃−2L₂, also, the fourth mounting arm hasthe inner side surface including the second width substantially equal tothe first width (W_(H)+2W_(S)), therefore, when each of the mountingarms 36, 37 has the inner side surface including the length of L₃−2L₂,the third mounting arm has the inner side surface including the firstwidth (W_(H)+2W_(S)) and the fourth mounting arm has the inner sidesurface including the second width substantially equal to the firstwidth (W_(H)+2W_(S)), an overall length dimension of the inner sidesurface of each of the mounting arms 36, 37, and the third and fourthmounting arms have a second overall length dimension which is given as2(L₃−2L₂+W_(H)+2W_(S)).

In order to obtain a reduced area of a mounting surface of the unitcomprising the resonator, the case and the lid, the first overall lengthdimension is greater than 2.4 mm and less than 5 mm, preferably, withina range of 2.5 mm to 4.8 mm, and the second overall length dimension isgreater than 1.6 mm and less than 4.4 mm, preferably, within a range of1.8 mm to 3.8 mm, as a result of which the unit comprising theresonator, the case and the lid can be obtained with a good aging, asmall capacitance ratio r, a small series resistance R₁ and strongstrength against a shock, namely, with a high electro-mechanicaltransformation efficiency even when the resonator of the unit isextremely miniaturized.

In other words, each of the mounting arms 36, 37 has first, second andthird outer side surfaces and each of the third and fourth mounting armshas a fourth outer side surface, the first outer side surface of each ofthe mounting arms 36, 37 has a first length, the second outer sidesurface of each of the mounting arms 36, 37 has a first width, the thirdouter side surface of each of the mounting arms 36, 37 has a secondwidth, and the fourth outer side surface of each of the third and fourthmounting arms has a third width, in addition, when the sum of the firstlength of the first outer side surface of each of the mounting arms 36,37, the first width of the second outer side surface of each of themounting arms 36, 37, the second width of the third outer side surfaceof each of the mounting arms 36, 37, and the third width of the fourthouter side surface of each of the third and fourth mounting arms isdefined by a first overall length dimension, the first overall lengthdimension is greater than 2.4 mm and less than 5 mm, preferably, withina range of 2.5 mm to 4.8 mm.

Similarly, each of the mounting arms 36, 37 has a first inner sidesurface opposite the first outer side surface, the third mounting armhas a second inner side surface, and the fourth mounting arm has a thirdinner side surface opposite the fourth outer side surface, also, thefirst inner side surface of each of the mounting arms 36, 37 has asecond length, and each of the second inner side surface of the thirdmounting arm and the third inner side surface of the fourth mounting armhas a fourth width, where the fourth width of the second inner sidesurface of the third mounting arm has the width of the base portion 40and the width of each of the first and second connecting portions 34,35, therefore, when the sum of the second length of the first inner sidesurface of each of the mounting arms 36, 37, and the fourth width ofeach of the second inner side surface of the third mounting arm and thethird inner side surface of the fourth mounting arm is defined by asecond overall length dimension, the second overall length dimension isgreater than 1.6 mm and less than 4.4 mm, preferably, within a range of1.8 mm to 3.8 mm, as a result of which the resonator 10 can be obtainedwith a small series resistance R₁, good aging behaviour, a smallcapacitance ratio r and strong strength against a shock, even when theresonator is extremely miniaturized.

As described below, when the base portion 40 has first and second baseportions and two cut portions located between the first and second baseportions, and each of the vibrational arms 20, 31 is connected to thefirst base portion of the base portion 40, the third frame portion whichhas the base portion 40 and the connecting portions 34, 35 comprises thesecond base portion of the base portion 40 and the connecting portions34, 35, and also a width of the second base portion of the base portion40 is equal to the width of the base portion 40.

In other words, as is obvious from FIG. 2 , the width of the second baseportion is equal to a width of the first base portion of the baseportion 40.

It is needless to say that each of the mounting arms 36, 37, and thethird and fourth mounting arms is integratedly formed with thevibrational arms 20, 31. In addition, the length of the fourth mountingarm is greater than or equal to each of a first spaced-apart distancebetween the fourth mounting arm and the vibrational arm 20 and a secondspaced-apart distance between the fourth mounting arm and thevibrational arm 31.

In other words, the first spaced-apart distance is a distance in thelength direction of the vibrational arm 20 measured from a first outeredge of the vibrational arm 20 to a first outer edge of the fourthmounting arm and the second spaced-apart distance is a distance in thelength direction of the vibrational arm 31 measured from a second outeredge of the vibrational arm 31 to a second outer edge of the fourthmounting arm, and each of the first spaced-apart distance and the secondspaced-apart distance is within a range of 0.015 mm to 0.2 mm,preferably, within a range of 0.02 mm to 0.16 mm, more preferably,within a range of 0.025 mm to 0.1 mm to get the vibrational arms 20, 31which vibrate easily and symmetrically in a flexural mode of an inversephase, as a result of which the resonator having the vibrational arms 2031 can be obtained with a good aging behaviour and a small seriesresistance R₁.

In addition, the width W₆ of each of the mounting arms 36, 37 isdifferent from or equal to the length of the fourth mounting arm,namely, the width W₆ is less or greater than the length of the fourthmounting arm. For example, the length of the fourth mounting arm isgreater than 0.03 mm, e.g. greater than 0.035 mm and less than 0.4 mm,preferably, within a range of 0.04 mm to 0.35 mm to get the fourthmounting arm with strong strength against a shock and to get asmall-sized package comprising a resonator with a high quality factor Q.

Also, a length of the first base portion of the base portion 40 isdifferent from or equal to the length of the fourth mounting arm,namely, the length of the first base portion is less than or greaterthan the length of the fourth mounting arm to get the resonator withstrong strength against a shock.

In addition, it is needless to say that the vibrational arms 20, 31shown in FIG. 2 may have vibrational arms 131, 132 shown in FIG. 10 , ashape of which is different from that of the vibrational arms 20, 31,namely, the vibrational arm 131 is replaced with the vibrational arm 20and the vibrational arm 132 is replaced with the vibrational arm 31,therefore, each of the vibrational arms 20, 31 comprises a plurality ofvibrational portions having a first vibrational portion including afirst width W and a first length within a range of 0.32 mm to 0.92 mm,preferably, within a range of 0.32 mm to 0.72 mm, more preferably,within a range of 0.32 mm to 0.53 mm, and a second vibrational portionincluding a second width We greater than the first width W and a secondlength less than the first length, and a groove is formed in each ofupper and lower surfaces of the first vibrational portion of each of thevibrational arms 20, 31. Also, the second width of the secondvibrational portion of each of the vibrational arms 20, 31 is greaterthan or equal to a spaced-apart distance between the first vibrationalportion of the vibrational arm 20 and the first vibrational portion ofthe vibrational arm 31.

In this embodiment, though the second width of the second vibrationalportion of each of the vibrational arms 20, 31 is greater than thespaced-apart distance between the first vibrational portion of thevibrational arm 20 and the first vibrational portion of the vibrationalarm 31, the second width of the second vibrational portion of each ofthe vibrational arms 20, 31 may be less than the spaced-apart distancebetween the first vibrational portion of the vibrational arm 20 and thefirst vibrational portion of the vibrational arm 31 and greater than aspaced-apart distance between the second vibrational portion of thevibrational arm 20 and the second vibrational portion of the vibrationalarm 31 which is within a range of 0.042 mm to 0.062 mm.

Namely, as a ratio of a mass of the second vibrational portion and amass of the first vibrational portion becomes very large, the resonatorhaving the vibrational arms 20, 31 each including the first and secondvibrational portions can be obtained with good aging behaviour, a smallseries resistance R₁ and strong strength against a shock.

In addition, when a spaced-apart distance between the second vibrationalportion of the vibrational arm 20 and the mounting arm 36 is defined bya first spaced-apart distance and a spaced-apart distance between thesecond vibrational portion of the vibrational arm 31 and the mountingarm 37 is defined by a second spaced-apart distance, each of the firstand second spaced-apart distances is different from or equal to aspaced-apart distance between the second vibrational portion of thevibrational arm 20 and the second vibrational portion of the vibrationalarm 31, and is within a range of 0.012 mm to 0.15 mm, namely, each ofthe first and second spaced-apart distances is greater than or less thanthe spaced-apart distance between the second vibrational portion of thevibrational arm 20 and the second vibrational portion of the vibrationalarm 31.

Also, when a spaced-apart distance between the first base portion of thebase portion and the first mounting arm is defined by a thirdspaced-apart distance and a spaced-apart distance between the first baseportion of the base portion and the second mounting arm is defined by afourth spaced-apart distance, each of the third and fourth spaced-apartdistances is greater than each of the first and second spaced-apartdistances.

However, this invention is not limited to this, but includes theplurality of vibrational portions having a third vibrational portionincluding a third width Wm greater than the first width W and less thanthe second width We, and a third length less than each of the first andsecond lengths, and the third vibrational portion is located between thefirst and second vibrational portions to prevent leakage of vibrationwhich is caused by vibration and to get the vibrational arms 20, 31which vibrate easily and symmetrically in a flexural mode of an inversephase, and also a ratio (We/W) of the second width We of the secondvibrational portion of each of the vibrational arms 20, 31 and the firstwidth W of the first vibrational portion of the corresponding one of thevibrational arms 20, 31 is within a range of 1.8 to 3.6, preferably,within a range of 2.1 to 2.9 or within a range of 2.75 to 3.4, morepreferably, within a range of 3.3 to 6.1 or within a range of 3.5 to5.9.

In addition, a ratio (Wm/W) of the third width Wm of the thirdvibrational portion of each of the vibrational arms 20, 31 and the firstwidth W of the first vibrational portion of the corresponding one of thevibrational arms 20, 31 is within a range of 1.6 to 2.3. By providingabove-described relations, a resonator can be obtained with good agingbehaviour because it has no vibration leakage which is caused byvibration.

Also, when a spaced-apart distance between the second vibrationalportion of the vibrational arm 20 and the second vibrational portion ofthe vibrational arm 31 is defined by a first spaced-apart distance and aspaced-apart distance between the second vibrational portion of thevibrational arm 20 and the mounting arm 36 having the length L₃ and thewidth W₆ is defined by a second spaced-apart distance, and aspaced-apart distance between the second vibrational portion of thevibrational arm 31 and the second mounting arm 37 having the length L₃and the width W₆ is defined by a third spaced-apart distance, a ratio ofthe first spaced-apart distance and the second spaced-apart distance orthe third spaced-apart distance is within a range of 0.45 to 2.22,preferably, within a range of 0.75 to 1.33, as a result of which thevibrational arms 20, 31 vibrate easily and symmetrically in a flexuralmode of an inverse phase so that the resonator 10 has a small seriesresistance R₁ and good aging behaviour.

Furthermore, as shown in FIG. 2 , each of the mounting arms 36, 37 isformed substantially parallel to the vibrational arms 20, 31 and has amounting arm portion including a constant width W₆ and a length of L₃−L₁so that the length of L₃−L₁ is less than a length L of each of thevibrational arms 20, 31, where L₁ represents the length of the baseportion and L₃ represents the length of each of the mounting arms 36,37, when a case has first and second mounting portions, the mounting armportion of the mounting arm 36 is mounted on the first mounting portionof the case and the mounting arm portion of the mounting arm 37 ismounted on the second mounting portion of the case, and also a lid isconnected to the case to cover an open end of the case.

In addition, each of the vibrational arms 20 and 31 has a first mainsurface and a second main surface opposite the first main surface andside surfaces, and the vibrational arms 20 and 31 have grooves 21 and27, respectively, each of which has stepped portions comprising a firststepped portion and a second stepped portion. Also, the resonator 10 hascutting angles θ_(y), θ_(x) and θ_(z) which are within a range of −20°to +20°, −25° to +25° and −18° to +18°, respectively, namely, a cuttingangle of the resonator is within a range of ZYlwt(−20° to)+20°/(−25°to)+25°/(−18° to +18°). In this embodiment, the quartz crystal tuningfork resonator can vibrate in a flexural mode of a fundamental mode ofan inverse phase, and which is one of a contour mode quartz crystalresonator.

In more detail, the groove 21 is constructed to include a portion of acentral linear line 41 of the arm 20, and the groove 27 is similarlyconstructed to include a portion of a central linear line 42 of the arm31. Each of the grooves 21 and 27 has a width W₂, and the width W₂including a portion of the central linear lines 41 and 42, is preferablebecause a large moment of inertia occurs at the arms 20 and 31 and thearms can vibrate in a flexural mode easily. As a result, the quartzcrystal tuning fork resonator capable of vibrating in a fundamental modecan be obtained with a small series resistance R₁ and a high qualityfactor Q₁.

In addition, when each of the vibrational arms 20 and 31 has part widthsW₁ and W₃, an arm width W of the arms 20 and 31 has a relationship ofW=W₁+W₂+W₃, and the part widths W₁ and W₃ are constructed so that W₁≥W₃or W₁<W₃. In addition, the width W₂ is constructed so that W₂≥W₁, W₃. Inthis embodiment, also, the grooves are constructed at the arms so that aratio W₂/W of the width W₂ and the arm width W is greater than 0.35 andless than 1, preferably, within a range of 0.35 to 0.95 and a ratio t₁/tis less than 0.79, where t₁ and t are a thickness of the groove and thevibrational arms, as shown in FIG. 3 , to obtain a very large moment ofinertia of the vibrational arms. That is, the quartz crystal tuning forkresonator, capable of vibrating in the fundamental mode, and having afrequency of high stability can be provided with a small seriesresistance R₁, a high quality factor Q₁ and a small capacitance ratio r₁because it has a very large electromechanical transformation efficiency.

Likewise, each of the vibrational arms 20 and 31 has a length L and eachof the grooves 21 and 27 has a length l₀ (not shown here). In thisembodiment, a ratio of the length l₀ and the length L is within a rangeof 0.3 to 0.8 to get a quartz crystal tuning fork resonator with seriesresistance R₁ of a fundamental mode of vibration smaller than seriesresistance R₂ of a second overtone mode of vibration. In other words,the length l₀ is within a range of 30% to 80% to the length L.

However, to get a small-sized resonator with a small series resistanceR₁ and a relationship of R₁<R₂, the ratio of the length l₀ and thelength L is within a range of 0.472 to 0.758. In general, the length l₀is within a range of 0.25 mm to 1.25 mm, preferably, within a range of0.3 mm to 0.79 mm or within a range of 0.45 mm to 1.25 mm, morepreferably, within a range of 0.3 mm to 0.52 mm. Also, when a pluralityof grooves are formed in at least one of upper and lower faces of thearms and divided in the length direction of the arms, the length l₀ is atotal length of the grooves.

In addition, the electrodes 25 and 26 are disposed on the side surfacesof the vibrational arm 20 and the electrode 23 is disposed on thesurface of the groove 21, which extends on the mounting arm 36 havingthe connecting portion 34. Namely, as already described above, theelectrode 23 of the groove 21 is connected to the electrode disposed onthe surface of the mounting arm 36 through the electrode disposed on thesurface of the connecting portion 34. Similar to this, the electrodes 32and 33 are disposed on the side surfaces of the vibrational arm 31 andthe electrode 29 is disposed on the surface of the groove 27, whichextends on the mounting arm 37 having the connecting portion 35. Namely,as already described above, the electrode 29 of the groove 27 isconnected to the electrode disposed on the surface of the mounting arm37 through the electrode disposed on the surface of the connectingportion 35. Each of the connecting portions 34 and 35 has the length L₂and the width W_(S), and each of the mounting arms has the length L₃ andthe width W₆. Also, the electrode 23 is connected to the electrodes 32and 33, and the electrode 29 is connected to the electrodes 25 and 26.

In this embodiment, the length l₀ of the groove corresponds to a lengthl_(d) of the electrode disposed inside each of the grooves, when thelength l_(d) of the electrode is less than the length l₀ of the groove,namely, the length l₀ is of the length l_(d) of the electrode. Inaddition, the base portion 40 has the length L₁ and the width W_(H), thelength L₁ is less than 0.5 mm, preferably, within a range of 0.015 mm to0.49 mm, more preferably, within a range of 0.12 mm to 0.45 mm, and atotal length L₁ (=L+L₁) in this embodiment is less than 2.1 mm,preferably, within a range of 0.6 mm to 1.0 mm or within a range of 0.8mm to 1.95 mm, more preferably, within a range of 1.02 mm to 1.95 mm toobtain a miniature quartz crystal resonator. Also, when a distance W₄between the vibrational arms is taken, a total width W₅ (=2W+W₄) is lessthan 0.53 mm, preferably, within a range of 0.15 mm to 0.52 mm, and thewidth W₅ is equal to or less than the W_(H) which is less than 0.55 mm,preferably, within a range of 0.15 mm to 0.53 mm.

In addition, the length L₃ is greater than or equal to the length L₂ andalso, the length L₁ is greater than or equal to the length L₂ or thelength L₁ is less than the length L₂. In actual, a value of L₁-L₂ iswithin a range of −0.1 mm to 0.32 min, preferably, within a range of 0.1mm to 0.32 mm or within a range of 0 mm to 0.3 mm, more preferably, 0mm, namely, L₁=L₂, especially, when W_(H) is greater than W₅, a distanceL₄ between an edge of the connecting portion and an outer edge of thevibrational arm is within a range of 0.012 mm to 0.38 mm. In detail, asshown in FIG. 2 , each of the connecting portions 34, 35 has an outeredge and each of the vibrational arms 20, 31 has an outer edge, a firstdistance in the length direction of the connecting portion 34 measuredfrom the outer edge of the connecting portion 34 to the outer edge ofthe vibrational arm 20 is defined by L₄, and a second distance in thelength direction of the connecting portion 35 measured from the outeredge of the connecting portion 35 to the outer edge of the vibrationalarm 31 is defined by L₄, each of the first distance L₄ and the seconddistance L₄ is within a range of 0.012 mm to 0.38 mm. In addition, thebase portion 40 has a plurality of portions having a first cut portionand a second cut portion (not shown here), each of the first and secondcut portions is cut into the base portion 40 from a side surface havingthe distance L₄ between the edge of the connecting portion and the outeredge of the vibrational arm. Namely, the first cut portion is cut intothe base portion 40 from the side surface having the distance L₄ betweenthe edge of the connecting portion 34 and the outer edge of thevibrational arm 20 and the second cut portion is cut into the baseportion 40 from the side surface having the distance L₄ between the edgeof the connecting portion 35 and the outer edge of the vibrational arm31. In more detail, the base portion 40 has a first side surface and asecond side surface opposite the first side surface, the first cutportion is cut into the base portion 40 from an arbitrary position ofthe first side surface of the base portion 40 and the second cut portionis cut into the base portion 40 from an arbitrary position of the secondside surface of the base portion 40, preferably, the first cut portionand the second cut portion are formed symmetrical to an central linearline (portion) of the base portion 40. Namely, the first cut portion isformed opposite the second cut portion in the width direction of thebase portion 40. It is needless to say that the vibrational arms 20, 31are connected to a side surface of the base portion 40 different fromeach of the first and second side surfaces of the base portion 40.

In other words, the base portions 40 has two cut portions, and a firstbase portion and a second base portion and the two cut portions areformed between the first and second base portions of the base portion40, namely, a third base portion is formed between the first and secondbase portions, each of the vibrational arms 20 and 31 is connected tothe first base portion of the base portion 40. In addition, the mountingarms 36 and 37 are connected to the base portion 40, also, the secondbase portion of the base portion 40 has a first side surface and asecond side surface opposite the first side surface and each of theconnecting portions 34 and 35 has a first end portion and a second endportion opposite the first end portion. In addition, the first endportion of the connecting portion 34 is connected to the first sidesurface of the second base portion of the base portion 40 and the secondend portion of the connecting portion 34 is connected to the mountingarm 36, and the first end portion of the connecting portion 35 isconnected to the second side surface of the second base portion of thebase portion 40 and the second end portion of the connecting portion 35is connected to the mounting arm 37 so that the second base portion ofthe base portion 40 and the connecting portions 34 and 35 have a U-shapeor a concave shape with the mounting arms 36 and 37.

As is shown in FIG. 2 , each of the mounting arms 36 and 37 extendssubstantially parallel to the vibrational arms 20 and 31. However, thepresent invention is not limited to this, but includes the mounting arms36 and 37 each having at least one arm portion extending not parallel tothe vibrational arms 20 and 31, e.g., each of the mounting arms 36 and37 has a plurality of arm portions including first, second, third,fourth and fifth arm portions.

In detail, each of the first, third and fifth arm portions of each ofthe mounting arms 36 and 37 extends substantially parallel to thevibrational arms 20 and 31, and besides the first arm portion isconnected to the third arm portion through the second arm portionextending not parallel to the vibrational arms 20 and 31 and the thirdarm portion is connected to the fifth arm portion through the fourth armportion extending not parallel to the vibrational arms 20 and 31 so thata direction of the second arm portion extending not parallel to thevibrational arms 20 and 31 is different from that of the fourth armportion extending not parallel to the vibrational arms 20 and 31.

In addition, the fifth arm portion of each of the mounting arms 36 and37 is mounted on a mounting portion of a case. In more detail, the fiftharm portion has a plurality of surfaces including a first surface, asecond surface and a third surface, and each of the fist, second andthird surfaces of the fifth arm portion is mounted on a surface of themounting portion of the case by a conductive adhesive.

In this embodiment, accordingly, the first end portion of the connectingportion 34 is connected to the first side surface of the second baseportion of the base portion 40 and the second end portion of theconnecting portion 34 is connected to the first arm portion of themounting arm 36, and the first end portion of the connecting portion 35is connected to the second side surface of the second base portion ofthe base portion 40 and the second end portion of the connecting portion35 is connected to the first arm portion of the mounting arm 37 so thatthe second base portion of the base portion 40 and the connectingportions 34 and 35 have a U-shape or a concave shape with the first armportions of the mounting arms 36 and 37.

As described above, a part or all of each of the mounting arms 36 and 37extends substantially parallel to the vibrational arms 20 and 31, andbesides, the mounting arm 36 is connected to the second base portion ofthe base portion 40 through the connecting portion 34 and the mountingarm 37 is connected to the second base portion of the base portion 40through the connecting portion 35 so that the second base portion of thebase portion 40 and the connecting portions 34 and 35 have a U-shape ora concave shape with the parts or all of the mounting arms 36 and 37. Itis, therefore, obvious from this that the connecting portion 34 and thefirst arm portion of the mounting arm 36 have a L-shape in a top view ofFIG. 2 , and also, the connecting portion 35 and the first arm portionof the mounting arm 37 have a L-shape in a bottom view of FIG. 2 .

Moreover, the second base portion of the base portion 40 has aconnecting portion through which the quartz crystal tuning forkresonator is connected to a quartz crystal wafer and the quartz crystaltuning fork resonator is chipped (cut off) from the quartz crystal waferat the connecting portion of the second base portion of the base portion40. Also, the second base portion of the base portion 40 has a pluralityof different length including a first length which is less than or equalto a length of the first base portion of the base portion 40, and asecond length greater than the first length, and the connecting portionof the second base portion of the base portion 40 is connected to aportion having the first length.

In addition, the fifth arm portion of each of the mounting arms 36 and37 is chamfered so that the fifth arm portion of each of the mountingarms 36 and 37 has a plurality of different widths including a firstwidth and a second width greater than the first width, and a portionwhich has the second width of the fifth arm portion of each of themounting arms 36 and 37 is mounted on the mounting portion of the caseby a conductive adhesive.

In this embodiment, each of the mounting arms 36 and 37 has five armportions, but, may have at least three arm portions including a firstarm portion connected to the quartz crystal tuning fork base, a secondarm portion extending not parallel to the vibrational arms 20 and 30,and a third arm portion connected to the first arm portion through thesecond arm portion, a length of the third arm portion of each of thevibrational arms 20 and 31 is greater than or equal to a length of thefirst arm portion of the corresponding one of the vibrational arms 20and 31 and the third arm portion of each of the mounting arms 36 and 37is mounted on a mounting portion of a case through a conductiveadhesive.

In addition to this, when a spaced-apart distance between the third baseportion of the base portion 40 and the first arm portion of the mountingarm 36 is defined by a first distance and a spaced-apart distancebetween the vibrational arm 20 and the third arm portion of the mountingarm 36 is defined by a second distance, the first distance is greaterthan the second distance, while when a spaced-apart distance between thethird base portion of the base portion 40 and the first arm portion ofthe mounting arm 37 is defined by a third distance and a spaced-apartdistance between the vibrational arm 37 and the third arm portion of themounting arm 37 is defined by a fourth distance, the third distance isgreater than the fourth distance.

As a result of which, the quartz crystal tuning fork resonator can beobtained with a reduced energy leakage which is caused by vibration anda low series resistance R₁.

In addition, each of the vibrational arms 20 and 31 comprises at leasttwo vibrational portions having a first vibrational portion including agenerally tapered shape comprised of a plurality of different widthshaving a first width and a second width less than the first width, and asecond vibrational portion including a third width less than or equal tothe first width.

Needless to say, by replacing “first” with “second” and “second” with“first”, the above-described sentences can be expressed as follows,namely, each of the vibrational arms 20 and 31 comprises at least twovibrational portions having a first vibrational portion including agenerally tapered shape comprised of a plurality of different widthshaving a first width and a second width greater than the first width,and a second vibrational portion including a third width less than orequal to the second width, e.g., the first width of the firstvibrational portion is greater than 0.05 mm and less than 0.057 mm, andthe second width of the first vibrational portion is greater than 0.058mm and less than 0.1 mm, and when the first vibrational portion has afirst length, the second vibrational portion has a second length lessthan the first length.

In detail, the first vibrational portion of each of the vibrational arms20 and 31 has a first main surface and a second main surface oppositethe first main surface and a groove is formed in each of the first andsecond main surfaces of the first vibrational portion of each of thevibrational arms 20 and 31 so that a distance in the width direction ofthe groove measured from an outer edge of the groove to an outer edge ofthe corresponding one of the vibrational arms 20 and 31 is less than0.015 mm.

As shown in FIG. 2 , the groove of the vibrational arms 20 and 31 isactually formed in a central liner portion, and a length of the grooveis within a range of 0.72 mm to 0.785 mm.

Also, the groove formed in each of the first and second main surfaces ofthe first vibrational portion of each of the vibrational arms 20 and 31has a first outer edge opposite a first outer edge of the correspondingone of the vibrational arms 20 and 31 in the width direction and asecond outer edge opposite a second outer edge of the corresponding oneof the vibrational arms 20 and 31 in the width direction, the firstouter edge of the groove is not opposite the second outer edge of thegroove in the width direction, and a distance in the width direction ofthe groove measured from the first outer edge of the groove to the firstouter edge of the corresponding one of the vibrational arms 20 and 31 isdifferent from or equal to a distance in the width direction of thegroove measured from the second outer edge of the groove to the secondouter edge of the corresponding one of the vibrational arms 20 and 31.

In addition, the second vibrational portion of each of the vibrationalarms 20 and 31 has third and fourth opposite main surfaces, and a metalfilm for adjusting an oscillation frequency of the quartz crystal tuningfork resonator is disposed on at least one of the third and fourthopposite main surfaces of the second vibrational portion of each of thevibrational arms 20 and 31. Moreover, the second vibrational portion ofeach of the vibrational arms 20 and 31 may have a generally taperedshape or a constant width in the length direction, namely, not variablewidth. Also, the width W₆ is less than 0.45 mm, preferably, within arange of 0.025 mm to 0.4 mm, more preferably, within a range of 0.08 mmto 0.4 mm, In addition, the width W₆ is greater than the width W of thevibrational arms 20, 31, namely, the width W₆ is different from thewidth W and the width W₆ is, preferably, within a range of 1.1 times ofthe width W to 3.8 times of the width W to get the quartz crystal tuningfork resonator with the enough resistance against a shock.

However, this invention is not limited to this, but may include arelationship of the width W₆ less than or equal to the width W, and thelength L₃ is less than 2.1 mm, preferably, within a range of 0.3 mm to1.85 mm to reduce a leakage energy by vibration, and also, the widthW_(S) is less than 0.41 mm, preferably, within a range of 0.015 mm to0.14 mm, and the length L₂ is greater than 0.025 mm and less than 0.55mm, preferably, within a range of 0.04 mm to 0.5 mm to get a shock proofquartz crystal resonator having a reduced leakage energy by vibration.In addition, the length L₂ is less than the width W₆ to get the shockproof quartz crystal resonator having the reduced leakage energy byvibration.

In more detail, the width W_(H) of the base portion 40 is within a rangeof 0.15 mm to 0.53 mm or within a range of 0.21 mm to 0.38 mm, the widthW_(S) of each of the connecting portions 34, 35 is within a range of0.015 mm to 0.14 mm, a length of each of the connecting portions 34, 35is greater than or equal to the width W_(S) and is within a range of0.02 mm to 0.18 mm, the width W₆ of each of the mounting arms 36, 37 iswithin a range of 0.025 mm to 0.4 mm, and the length L₃ of each of themounting arms 36, 37 is within a range of 0.3 mm to 1.85 mm. Therefore,as minimum values of the width W_(H), the width W_(S), the width W₆ andthe length L₃ are given as 0.15 mm of the width W_(H), 0.015 mm of thewidth W_(S) of the connecting portion 34, 0.015 mm of the width W_(S) ofthe connecting portion 35, 0.025 mm of the width W₆ of the mounting arm36, 0.025 mm of the width W₆ of the mounting arm 37, 0.3 mm of thelength L₃ of the mounting arm 36, and 0.3 mm of the length L₃ of themounting arm 37, the sum of the minimum values is 0.83 mm.

Similar to this, as maximum values of the width W_(H), the width W_(S),the width W₆ and the length L₃ are given as 0.53 mm of the width W_(H),0.14 mm of the width W_(S) of the connecting portion 34, 0.14 mm of thewidth W_(S) of the connecting portion 35, 0.4 mm of the width W₆ of themounting arm 36, 0.4 mm of the width W₆ of the mounting arm 37, 1.85 mmof the length L₃ of the mounting arm 36, and 1.85 mm of the length L₃ ofthe mounting arm 37, the sum of the maximum values is 5.31 mm.

In other words, each of the base portion 40, the connecting portions 34,35, and the mounting arms 36, 37 has an outer side surface including awidth and each of the mounting arms 36, 37 has an outer side surfaceincluding a length, namely, when the sum of the width of the outer sidesurface of each of the base portion 40, the connecting portions 34, 35,and the mounting arms 36, 37 and the length of the outer side surface ofeach of the mounting arms 36, 37 is defined by an overall lengthdimension, the overall length dimension is within a range of 0.83 mm to5.31 mm.

Moreover, the distance W₄ and the width W₂ are constructed so thatW₄≥W₂, and more, the distance W₄ is within a range of 0.045 mm to 0.65mm and the width W₂ is within a range of 0.02 mm to 0.12 mm, preferably,within a range of 0.02 mm to 0.04 mm, more preferably, within a range of0.02 mm to 0.035 mm because it is easy to form a very small-sizedresonator shape and grooves formed at the vibrational arms separately bya photo-lithographic process and an etching process, consequently, afrequency stability for a fundamental mode of vibration of the resonatorgets higher than that for a second overtone mode of vibration thereof.In addition, the distance W₄ has a plurality of different spaced-apartdistances including a first spaced-apart distance comprised of agenerally tapered spaced-apart distance and a second spaced-apartdistance greater than the first spaced-apart distance. In thisembodiment, a quartz wafer having a thickness t of 0.045 mm to 0.35 mmis used.

For example, in order to get a smaller-sized quartz crystal tuning forkresonator, capable of vibrating in a flexural mode, it is necessary thatthe width W₂ of the groove is less than 0.07 mm, in addition to theabove-described relations, preferably, within a range of 0.01 mm to 0.12mm, more preferably, within a range of 0.012 mm to 0.025 mm or within arange of 0.038 mm to 0.05 mm, and the arm width W is less than 0.18 mm,and preferably, the W is greater than 0.025 mm and less than 0.175 mm,more preferably, greater than 0.025 mm and less than 0.1 mm or greaterthan 0.05 mm and less than 0.1 mm, and the arm length L is less than1.56 mm, preferably, within a range of 0.53 mm to 1.46 mm, morepreferably, within a range of 0.53 mm to 0.72 mm or within a range of0.56 mm to 1.15 mm. Also, the thickness t₁ of the groove is within arange of 0.01 mm to 0.085 mm approximately, and the part widths W₁ andW₃ are less than 0.021 mm, respectively, preferably, less than 0.015 mm,more preferably, within a range of 0.001 mm to 0.012 mm or within arange of 0.0025 mm to 0.0045 mm or within a range of 0.006 mm to 0.012mm, still more preferably, within a range of 0.001 mm to 0.003 mm orwithin a range of 0.0041 mm to 0.0062 mm. In addition, a groove providedon at least one of an obverse face and a reverse face of the vibrationalarms of this embodiment may be a through hole, namely, the thickness ofthe hole t₁=0.

In more detail, to obtain a quartz crystal tuning fork resonator,capable of vibrating in a flexural mode, and having a frequency of highstability which achieves high time accuracy, it is necessary to obtainthe resonator whose resonance frequency is not influenced by shuntcapacitance because quartz crystal is a piezoelectric material and thestability for frequency is very dependent on the shunt capacitance. Inorder to decrease the influence on the resonance frequency by the shuntcapacitance, a merit value M_(i) plays an important role. Namely, themerit value M_(i) that expresses inductive characteristics, anelectromechanical transformation efficiency and a quality factor of aquartz crystal tuning fork resonator, is defined by a ratio Q_(i)/r_(i)of a quality factor Q_(i) and capacitance ratio r_(i), namely, M_(i) isgiven by M_(i)=Q_(i)/r_(i), where i shows a vibration order of theresonator, and for example, when i=1 and 2, merit values M₁ and M₂ arefor a fundamental mode of vibration of the resonator and a secondovertone mode of vibration thereof, respectively.

Also, a frequency difference Δf of resonance frequency f_(s) ofmechanical series independent on the shunt capacitance and resonancefrequency f_(r) dependent on the shunt capacitance is inverselyproportional to the merit value M_(i). The larger the value M_(i)becomes, the smaller the difference Δf becomes. Namely, the influence onthe resonance frequency f_(r) by the shunt capacitance decreases becauseit is close to the resonance frequency f_(s). Accordingly, the largerthe M_(i) becomes, the higher the stability for frequency of the quartzcrystal tuning fork resonator becomes because the resonance frequencyf_(r) of the resonator is almost never dependent on the shuntcapacitance. Namely, the quartz crystal tuning fork resonator can beprovided with a high time accuracy.

In detail, the quartz crystal tuning fork resonator can be obtained withthe merit value M₁ of the fundamental mode of vibration greater than themerit value M₂ of the second overtone mode of vibration by providing theabove-described tuning fork shape, grooves and dimensions. That is tosay, a relationship of M₁>M₂ is obtained. As an example, when aresonance frequency of a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is about 32.768 kHz for a fundamental modeof vibration and the resonator has a value of W₂/W=0.5, t₁/t=0.34 andl₁/l=0.48, though there is a distribution in production, the resonatorhas a merit value of M₁>65 for the fundamental mode of vibration and amerit value of M₂<30 for the second overtone mode of vibration,respectively.

Namely, the quartz crystal tuning fork resonator can be provided withhigh inductive characteristics, good electromechanical transformationefficiency (small capacitance ratio r₁ and small series resistance R₁)and a high quality factor. As a result, a stability for frequency of thefundamental mode of vibration becomes higher than that of the secondovertone mode of vibration, and simultaneously, the second overtone modeof vibration can be suppressed because capacitance ratio r₂ and seriesresistance R₂ of the second overtone mode of vibration become greaterthan capacitance ratio r₁ and series resistance R₁ of the fundamentalmode of vibration, respectively. In particular, r₂ has a value greaterthan 1500 in this embodiment. In order to ensure the suppression of thesecond overtone mode of vibration, r₂ is, preferably, greater than 1800,more preferably, greater than 2000.

Therefore, the resonator capable of vibrating in the fundamental modevibration can be provided with a high time accuracy because it has thefrequency of high stability. Consequently, a quartz crystal oscillatorcomprising the quartz crystal tuning fork resonator of this embodimentoutputs an oscillation frequency of the fundamental mode vibration as anoutput signal, and the frequency of the output signal has a very highstability, namely, excellent time accuracy. In other words, the quartzcrystal oscillator of this embodiment has a remarkable effect such thata frequency change by ageing becomes extremely small. Also, anoscillation frequency of the quartz crystal resonator of this embodimentis adjusted so that a frequency deviation is within a range of −100 PPMto +100 PPM to a nominal frequency, e.g. 32.768 kHz, after mounting iton a mounting portion of a case for housing the resonator.

In addition, the groove thickness t₁, shown in FIG. 3 , of the presentinvention is the thinnest thickness of the grooves because quartzcrystal is an anisotropic material and the groove thickness t₁ has adistribution when it is formed by a chemical etching method. In detail,a groove shape of the sectional view of vibrational arms in FIG. 3 has arectangular shape, but the groove shape has an about U shape actually.In the above-described embodiments, though the grooves are constructedat the arms, this invention is not limited to this, namely, arelationship of the merit values M₁ and M₂ can be applied to theconventional quartz crystal tuning fork resonator and a relationship ofa quartz crystal oscillating circuit comprising an amplification circuitand a feedback circuit can be also applied to the conventional quartzcrystal tuning fork resonator to suppress a second overtone modevibration and to get a high frequency stability for a fundamental modeof vibration of the quartz crystal tuning fork resonator.

FIG. 3 shows an A-A′ cross-sectional view of the vibrational arms 20 and31 of the quartz crystal resonator 10 in FIG. 2 , and electrodeconstruction within the grooves. The vibrational arm 20 has grooves 21and 22 cut into it, which include a portion of central linear line ofthe arm 20. The grooves 21 and 22 have a first set of electrodes 23 and24 of the same electrical polarity, while the side surfaces of the arm20 have a second set of electrodes 25 and 26 having an oppositeelectrical polarity to the first set of electrodes 23 and 24. Thevibrational arm 31 has grooves 27 and 28 constructed in a similar manneras the vibrational arm 20. The grooves 27 and 28 have a third set ofelectrodes 29 and 30 of the same electrical polarity, and the sidesurfaces of the vibrational arm 31 have a fourth set of electrodes 32and 33 with the opposite electrical polarity to the third electrodes 29and 30. The electrodes disposed on the vibrational arms 20 and 31 areconnected as shown in FIG. 3 , namely, two electrode terminals ofdifferent electrical polarity C-C′ are obtained.

In detail, the first set of electrodes 23 and 24 disposed on the grooves21 and 22 of the vibrational arm 20 have the same electrical polarity asthe fourth set of electrodes 32 and 33 disposed on both side surfaces ofthe vibrational arm 31, while the second set of electrodes 25 and 26disposed on both side surfaces of the vibrational arm 20 have the sameelectrical polarity as the third set of electrodes 29 and 30 disposed onthe grooves 27 and 28 of the arm 31. When a direct voltage is appliedbetween the electrode terminals C-C′, an electric field Ex occurs alongthe arrow direction inside the vibrational arms 20 and 31. As theelectric field Ex occurs perpendicular to the electrodes disposed on thevibrational arms, as shown in the arrow signs, the electric field Ex hasa very large value and a large distortion occurs at the vibrationalarms. As a result, a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is obtained with a small series resistanceR₁ and a high quality factor Q because even when miniaturized there is avery large electromechanical transformation efficiency for theresonator.

FIG. 4 shows an A-A′ cross-sectional view of another embodiment of thevibrational arms 20 and 31 of the quartz crystal resonator 10 in FIG. 2. The vibrational arm 20 has a through groove 21 a, which include aportion of central linear line of the arm 20. The through groove 21 ahas a first set of electrodes 23 a and 24 a of the same electricalpolarity, while the side surfaces of the arm 20 have a second set ofelectrodes 25 and 26 having an opposite electrical polarity to the firstset of electrodes 23 a and 24 a. The vibrational arm 31 has a throughgroove 27 a constructed in a similar manner as the vibrational arm 20.The through groove 27 a has a third set of electrodes 29 a and 30 a ofthe same electrical polarity, and the side surfaces of the vibrationalarm 31 have a fourth set of electrodes 32 and 33 with the oppositeelectrical polarity to the third electrodes 29 a and 30 a. Theelectrodes disposed on the vibrational arms 20 and 31 are connected asshown in FIG. 4 , namely, two electrode terminals of differentelectrical polarity E-E′ are obtained.

In detail, the first set of electrodes 23 a and 24 a disposed on thethrough groove 21 a of the vibrational arm 20 have the same electricalpolarity as the fourth set of electrodes 32 and 33 disposed on both sidesurfaces of the vibrational arm 31, while the second set of electrodes25 and 26 disposed on both side surfaces of the vibrational arm 20 havethe same electrical polarity as the third set of electrodes 29 a and 30a disposed on the through groove 27 a of the arm 31. When a directvoltage is applied between the electrode terminals E-E′, an electricfield Ex occurs along the arrow direction inside the vibrational arms 20and 31. As the electric field Ex occurs perpendicular to the electrodesdisposed on the vibrational arms, as shown in the arrow signs, theelectric field Ex has a very large value and a large distortion occursat the vibrational arms. As a result, a quartz crystal tuning forkresonator capable of vibrating in a flexural mode is obtained with asmall series resistance R₁ and a high quality factor Q because even whenminiaturized there is a very large electromechanical transformationefficiency for the resonator.

FIG. 5 shows a plan view of a quartz crystal resonator 50 of a secondembodiment of the present invention, and which is a quartz crystaltuning fork resonator capable of vibrating in a flexural mode. Theresonator 50 comprises vibrational arms 60 and 71 and a base portion 80attached to the vibrational arms, and the base portion 80 has a mountingarm 77 protruding from the base portion, and the mounting arm 77 ismounted on a mounting portion of a package comprising a case for housingthe resonator and a lid for covering an open end of the case. Inaddition, each of the vibrational arms 60 and 71 has a first mainsurface and a second main surface and side surfaces, and the vibrationalarms 60 and 71 have grooves 61 and 67, respectively, each of which hasstepped portions comprising a first stepped portion and a second steppedportion. Also, the resonator 50 has the same cutting angles θ_(y), θ_(x)and θ_(z) and the same dimensions W₁, W₂, W₃, W₄, W₅, W, L₁, L₂, L₃, L₄and Las the resonator of FIG. 2 .

In more detail, the groove 61 is constructed to include a portion of acentral linear line 81 of the arm 60, and the groove 67 is similarlyconstructed to include a portion of a central linear line 82 of the arm71. Each of the grooves 61 and 67 has a width W₂, and the width W₂includes a portion of the central linear lines 81 and 82 because a largemoment of inertia occurs at the arms 60 and 71. In this embodiment, theresonator is a quartz crystal tuning fork resonator capable of vibratingin a flexural mode and which can vibrate in a fundamental mode of aninverse phase easily. As a result, the quartz crystal tuning forkresonator capable of vibrating in a fundamental mode of an inverse phasecan be obtained with a small series resistance R₁ and a high qualityfactor Q₁.

In addition, electrodes 65 and 66 are disposed on side surfaces of thevibrational arm 60 and an electrode 63 is disposed on a surface of thegroove 61, and which is connected to electrodes 72 and 73 disposed onside surfaces of the vibrational arm 71. Similar to this, an electrode69 is disposed on a surface of the groove 67, which extends on themounting arm 77 having a first width W₆ and a second width greater thefirst width W₆ and a connecting portion 75. Namely, the mounting arm 77has a first arm portion including the first width W₆ and a second armportion including the second width greater than the first width W₆ inFIG. 5 , and the second arm portion of the mounting arm 77 is mounted ona mounting portion 177 of a case 175 in FIG. 18 . Namely, the mountingarm 77 is connected to the base portion 80 through the connectingportion 75 so that the mounting arm 77 extends in a common directionwith the vibrational arms 60, 71 outside the vibrational arms 60, 71, asshown in FIG. 5 . In detail, an electrode is disposed on a surface ofthe groove formed in a central linear portion of each of the first andsecond main surfaces of each of the vibrational arms 60, 71, and fromFIG. 5 , the electrode disposed on the surface of the groove formed inthe central Linear portion of at least one of the first and second mainsurfaces of one of the vibrational arms 60, 71 is connected to anelectrode disposed on a surface of the mounting arm 77 through aelectrode disposed on a surface of the connecting portion 75, and theelectrode disposed on the surface of the mounting arm 77 is connected toan electrode disposed on a surface of the mounting portion 177 of thecase 175 as shown in FIG. 17 , needless to say, the mounting arm 77 hasan electrode disposed on a surface of each of the first and second armportions of the mounting arm 77 as shown in FIG. 5 , and the electrodedisposed on the surface of the second arm portion of the mounting arm 77is connected to the electrode disposed on the surface of the mountingportion 177 of the case 175 as shown in FIG. 17 . In addition, each ofthe first and second arm portions of the mounting arm has a length,though the length of the second arm portion of the mounting arm is lessthan the length of the first arm portion thereof in this embodiment,However, the present invention is not limited to this, but, includes thelength of the second arm portion of the mounting arm greater than thelength of the first arm portion thereof. Also, the electrode 69 isconnected to the electrodes 65 and 66.

FIG. 6 shows a plan view of a quartz crystal resonator 90 of a thirdembodiment of the present invention, which is a quartz crystal tuningfork resonator capable of vibrating in a flexural mode. The resonator 90comprises vibrational arms 91 and 92 and a base portion 95 attached tothe vibrational arms, and the base portion 90 has a mounting arm 96protruding from the base portion. In this embodiment, the mounting arm96 is between the vibrational arms 91 and 92 and is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In addition,each of the vibrational arms 91 and 92 has a first main surface and asecond main surface opposite the first main surface and side surfaces,and the vibrational arms 91 and 92 have grooves 93 and 94, respectively,each of which has stepped portions comprising a first stepped portionand a second stepped portion. Also, the resonator 90 has the samecutting angles θ_(y), θ_(x) and θ_(z) as the quartz crystal resonator 10of FIG. 2 . In this embodiment, the quartz crystal tuning fork resonatorcan vibrate in a flexural mode of a fundamental mode of an inversephase.

In detail, similar to the resonator of FIG. 2 , the groove 93 is alsoconstructed to include a portion of a central linear line of the arm 91,and also, the groove 94 is similarly constructed to include a portion ofa central linear line of the arm 92. Each of the grooves 93 and 94 has awidth W₂, and the width W₂ includes a portion of the central linearlines because a large moment of inertia occurs at the vibrational arms91 and 92 and which can vibrate in a flexural mode easily. As a result,the quartz crystal tuning fork resonator capable of vibrating in afundamental mode can be obtained with a small series resistance R₁ and ahigh quality factor Q₁.

In addition, the base portion 95 has a length L₁ and a width W_(H).Also, each of the vibrational arms 91 and 92 has a width W, part widthsW₁ and W₃ and a width W₂ of the groove, namely, there is a relationshipof W=W₁+W₂+W₃. In detail, the resonator 90 has the same dimensions L₁,W_(H), W₁, W₂, W₃, and W and the same relationship as the resonator 10of FIG. 2 . Moreover, the mounting arm 96 has a width W₁₁ and there is adistance W₁₀ between the vibrational arm 91 or the vibrational arm 92and the mounting arm 96. In order to get a quartz crystal tuning forkresonator with reduced leakage energy by vibration of the vibrationalarms, the distance W₁₀ is within a range of 0.032 mm to 0.21 mm and thewidth W₁₁ is within a range of 0.21 mm to 0.88 mm. In addition, to get ashockproof quartz crystal tuning fork resonator, W₁₁ has a relationshipof (1.2 to 7.6)×W. In this embodiment, a total width W_(t)(=2W+2W₁₀+W₁₁)is less than 1.3 mm, preferably, within a range of 0.52 mm to 1.2 mm, toget a miniature quartz crystal tuning fork resonator.

FIG. 7 shows a B-B′ cross-sectional view of the vibrational arms 91 and92 of the resonator 90 in FIG. 6 . The vibrational arm 91 has grooves 93and 97 cut into it, and the grooves 93 and 97 have a first set ofelectrodes 100 and 101 of the same electrical polarity, while the sidesurfaces of the arm 91 have a second set of electrodes 99 and 102 havingan opposite electrical polarity to the first set of electrodes 100 and101. The vibrational arm 92 has grooves 94 and 98 constructed in asimilar manner as the vibrational arm 91. The grooves 94 and 98 have athird set of electrodes 106 and 107 of the same electrical polarity, andthe side surfaces of the vibrational arm 92 have a fourth set ofelectrodes 105 and 108 with the opposite electrical polarity to thethird electrodes 106 and 107. The electrodes disposed on the vibrationalarms 91 and 92 are connected as shown in FIG. 7 , namely, two electrodeterminals of different electrical polarity D-D′ are obtained.

In detail, the first set of electrodes 100 and 101 disposed on thegrooves 93 and 97 of the vibrational arm 91 have the same electricalpolarity as the fourth set of electrodes 105 and 108 disposed on bothside surfaces of the vibrational arm 92 and an electrode 104 disposed onthe mounting arm 96, while the second set of electrodes 99 and 102disposed on both side surfaces of the vibrational arm 91 have the sameelectrical polarity as the third set of electrodes 106 and 107 disposedon the grooves 94 and 98 of the arm 92 and an electrode 103 disposed onthe mounting arm 96. When a direct voltage is applied between theelectrode terminals D-D′, an electric field Ex occurs along the arrowdirection inside the vibrational arms 91 and 92. As the electric fieldEx occurs perpendicular to the electrodes disposed on the vibrationalarms, as shown in the arrow signs, the electric field Ex has a verylarge value and a large distortion occurs at the vibrational arms.Consequently, a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode is obtained with a small series resistanceR₁ and a high quality factor Q because even when miniaturized, there isvery large electromechanical transformation efficiency for theresonator.

FIG. 8 shows a plan view of a quartz crystal resonator 110 of a fourthembodiment of the present invention, which is a quartz crystal tuningfork resonator. The resonator 110 comprises vibrational arms 111 and 112and a base portion 113 attached to the vibrational arms, and the baseportion 113 has mounting arms 114 and 115 protruding from the baseportion, each of which is mounted on a mounting portion of a packagecomprising a case for housing the resonator and a lid for covering anopen end of the case. In this embodiment, the vibrational arms 111 and112 have grooves 116 and 117, respectively. In more detail, each of thevibrational arms 111 and 112 has an end portion connected to the baseportion and a free end portion, when a distance measured from the endportion to the free end portion is a length L, each of the vibrationalarms has a width W between the end portion and half a length L/2 of eachof the vibrational arms and a width W_(e) between half the length L/2and the length L of the free end portion of each of the vibrationalarms, and the width W is less than the width W_(e).

In other words, as shown in FIG. 8 , each of the vibrational arms 111and 112 comprises a first vibrational portion having a first width W anda first length, and a second vibrational portion having a second widthWe greater than the first width W and a second length less than thefirst length, and a portion protrudes from the second vibrationalportion of each of the vibrational arms 111 and 112 and extends in adirection of the base portion so that a spaced-apart distance is locatedbetween the first vibrational portion of each of the vibrational arms111 and 112 and the portion protruding from the second vibrationalportion of the corresponding one of the vibrational arms 111 and 112. Inthis embodiment, the quartz crystal tuning fork resonator can vibrate ina flexural mode and vibrate in a fundamental mode of an inverse phase.

However, this invention is not limited to this, but may have two groovesformed on at least one or each of an upper face (third main surface) anda lower face (fourth main surface opposite the third main surface) ofthe second vibration portion of each of the vibrational arms 111 and 112so that a length of each of the two grooves is within a range of 0.015mm to 0.095 mm or a through-hole formed at the second vibration portionof each of the vibrational arms 111 and 112.

Similar to this, a quartz crystal tuning fork resonator 120 of a fifthembodiment of the present invention is shown in FIG. 9 showing a planview thereof. The resonator 120 comprises vibrational arms 121 and 122and a base portion 123 attached to the vibrational arms, and the baseportion 123 has a mounting arm 124 protruding from the base portion. Inthis embodiment, the mounting arm 124 is between the vibrational arms121 and 122 and the vibrational arms 121 and 122 have grooves 125 and126, respectively. For this case the width W is also less than the widthW_(e) similar to that of FIG. 8 .

FIG. 10 shows a plan view of a quartz crystal resonator 130 of a sixthembodiment of the present invention, which is a quartz crystal tuningfork resonator. The resonator 130 comprises vibrational arms 131 and 132and a base portion 133 attached to the vibrational arms, and the baseportion 133 has mounting arms 134 and 135 protruding from the baseportion, each of which is mounted on a mounting portion of a packagecomprising a case for housing the resonator and a lid for covering anopen end of the case. In this embodiment, the vibrational arms 131 and132 have grooves 136 and 137, respectively.

In more detail, each of the vibrational arms 131 and 132 has an endportion connected to the base portion and a free end portion, when adistance measured from the end portion to the free end portion is alength L, each of the vibrational arms has a width W between the endportion and half a length L/2 of each of the vibrational arms and awidth W_(e) between half the length L/2 and the length L of the free endportion of each of the vibrational arms, and the width W is less thanthe width We. Namely, as is obvious from FIG. 10 , a length of a portionhaving the width We is less than a length of a portion having the widthW. Also, the resonator 130 similar to the resonator 10 in FIG. 2 has anoverall length less than 2.1 mm, preferably, within a range of 1.02 mmto 1.95 mm. In this embodiment, the quartz crystal tuning fork resonatorcan vibrate in a flexural mode and vibrate in a fundamental mode of aninverse phase.

In detail, as shown in FIG. 10 , each of the vibrational arms 131 and132 comprises a plurality of vibrational portions having a firstvibrational portion including a first length, e.g., within a range of0.32 mm to 0.92 mm, preferably, within a range of 0.32 mm to 0.72 mm,more preferably, within a range of 0.32 mm to 0.53 mm, and a first widthW, and a second vibrational portion including a second length, e.g.,within a range of 0.17 mm to 0.38 mm and less than the first length anda second width We greater than the first width W, and a groove 136 isformed in at least one of upper and lower faces of the first vibrationalportion of the vibrational arm 131 and a groove 137 is formed in atleast one of upper and lower faces of the first vibrational portion ofthe vibrational arm 132, and also a width of the groove formed in the atleast one of the upper and lower faces of each of the vibrational arms131, 132 is less than 0.07 mm, and a distance in the width direction ofthe groove measured from an outer edge of the groove to an outer edge ofthe corresponding one of the vibrational arms 131, 132 is less than0.015 mm as shown in FIG. 2 .

In addition, as shown in FIG. 16(a), each of the vibrational arms 131and 132 is connected to a base portion which has a first base portionincluding a first length and a first width, and a second base portionincluding a second length and a second width greater than the firstwidth, and each of the vibrational arms 131 and 132 is connected to thefirst base portion of the base portion, the second base portion ismounted on a mounting portion of a case. In addition to this, when aratio of the second width of the second vibrational portion of each ofthe vibrational arms 131, 132 and the first length of the first baseportion of the base portion is defined by a first ratio, the first ratiois within a range of 0.55 to 0.75.

In more detail, the second width We of the second vibrational portion ofeach of the vibrational arms 131, 132 is less than or equal to thesecond length of the second vibrational portion of the corresponding oneof the vibrational arms 131, 132, and the second width We is within arange of 0.056 mm to 0.26 mm, preferably, within a range of 0.056 mm to0.085 mm or within a range of 0.07 mm to 0.98 mm, more preferably,within a range of 0.12 mm to 0.26 mm and the second length is within arange of 0.17 mm to 0.38 mm. Also, the second width We of the secondvibrational portion of the vibrational arm 131 is greater than a firstspaced-apart distance between the first vibrational portion of thevibrational arm 131 and the mounting arm 134, and the second width We ofthe second vibrational portion of the vibrational arm 132 is greaterthan a second spaced-apart distance between the first vibrationalportion of the vibrational arm 132 and the mounting arm 135, and each ofthe first and second spaced-apart distances is within a range of 0.02 mmto 0.18 mm, preferably, within a range of 0.03 mm to 0.16 mm.

Also, a spaced-apart distance between the first vibrational portion ofthe vibrational arm 131 and the first vibrational portion of thevibrational arm 132 is greater than the first spaced-apart distancebetween the first vibrational portion of the vibrational arm 131 and themounting arm 134 and the second spaced-apart distance between the firstvibrational portion of the vibrational arm 132 and the mounting arm 135.

In addition, the second width We of the second vibrational portion ofeach of the vibrational arms 131, 132 is greater than a spaced-apartdistance between the second vibrational portion of the vibrational arm131 and the second vibrational portion of the vibrational arm 132, andthe spaced-apart distance is less than each of a spaced-apart distancebetween the mounting arm 134 and the first vibrational portion of thevibrational arm 131 and a spaced-apart distance between the mounting arm135 and the first vibrational portion of the vibrational arm 132 and iswithin a range of 0.015 mm to 0.07 mm.

However, this invention is not limited to this embodiment having thefirst and second vibrational portions, but includes a third vibrationalportion having a third length less than the first length of the firstvibrational portion and the second length of the second vibrationalportion and a third width greater than the first width W and less thanthe second width We, and the third vibrational portion is locatedbetween the first and second vibrational portions. In addition, the sumof the second length of the second vibrational portion of thevibrational arm 131 and the third length of the third vibrationalportion of the vibrational arm 131 is defined by a fourth length and thefourth length is less than the first length of the first vibrationalportion of the vibrational arm 131.

Similarly, the sum of the second length of the second vibrationalportion of the vibrational arm 132 and the third length of the thirdvibrational portion of the vibrational arm 132 is defined by a fifthlength and the fifth length is less than the first length of the firstvibrational portion of the vibrational arm 132.

In this embodiment, though the third vibrational portion is formedbetween the second vibrational portion and the first vibrationalportion, it is needless to say that the second vibrational portion andthe third vibrational portion may be newly replaced with a secondvibrational portion which has a first portion including the third widthand the third length, and a second portion including the second width Wegreater than the third width and the second length greater than thethird length. Namely, the first portion of the second vibrationalportion is formed between the second portion of the second vibrationalportion and the first vibrational portion of the corresponding one ofthe vibrational arms 131, 132.

In addition, the base portion 133 similar to the base portion 40 shownin FIG. 2 has a length less than 0.5 mm, preferably, within a range of0.015 mm to 0.49 mm and a width less than 0.55 mm, and first and secondbase portions and two cut portions located between the first and secondbase portions so that the second base portion has a width equal to thewidth of the base portion, and each of the vibrational arms 131, 132 isconnected to the first base portion of the base portion 133, namely, themounting arm 134 is connected to the second base portion of the baseportion 133 through a connecting portion and the mounting arm 135 isconnected to the second base portion of the base portion 133 through aconnecting portion so that each of the mounting arms 134, 135 extends ina common direction with the vibrational arms 131, 132 outside thevibrational arms 131, 132, and the mounting arms 134, 135 have a concaveshape or a U-shape with the second base portion of the base portion 133and the connecting portions.

In addition to this, a length of each of the connecting portions isgreater than a length of the first base portion of the base portion 133and greater than or equal to a width W_(S) (see FIG. 2 ) of each of theconnecting portions. As is obvious from FIG. 2 , it is needless to saythat the width of the second base portion of the base portion is equalto a width of the first base portion of the base portion because the twocut portions are formed between the first and second base portions andthe mounting arms 134, 135 are connected to the second base portion ofthe base portion through the connecting portions each having the widthW_(S). Namely, each of the first and second base portions has the widthW_(H).

Also, a length of each of the mounting arms 134, 135 is within a rangeof 0.3 mm to 1.85 mm and a width of each of the mounting arms 134, 135is less than 0.45 mm and a width of each of the connecting portions isless than 0.41 mm as shown in FIG. 2 . In addition, the second length ofthe second vibrational portion of each of the vibrational arms 131, 132is greater than a length of each of the first and second base portionsof the base portion 133.

In other words, the base portion 133 has a third base portion betweenthe first and second base portions, namely, connecting the first baseportion to the second base portion, and a width of the third baseportion is less than a width of each of the first and second baseportions, and a length of the first base portion is greater than orequal to a length of each of the second and third base portions and iswithin a range of 0.025 mm to 0.12 mm. Namely, the length of the secondand third base portions is less than or equal to the length of the firstbase portion. Also, each of the vibrational arms 131, 132 is connectedto the first base portion of the base portion 133. In addition, thewidth of the third base portion is greater than the length of the thirdbase portion, and greater than 0.04 mm and less than 0.2 mm, preferably,within a range of 0.045 mm to 0.18 mm and the length of the third baseportion is greater than 0.01 mm and less than 0.12 mm, preferably,within a range of 0.015 mm to 0.08 mm.

In addition to this, a ratio of the length of the third base portion andthe width of the third base portion is within a range of 0.45 to 0.487to get a resonator with no vibration leakage which is caused byvibration, high shock-proof properties and good aging characteristics.

In addition, the second length of the second vibrational portion of eachof the vibrational arms 131, 132 is greater than the length of the firstbase portion of the base portion 133, and is greater than 0.15 mm andless than 0.5 mm, preferably, within a range of 0.17 mm to 0.38 mm.

In addition to this, the second width We of the second vibrationalportion of each of the vibrational arms 131, 132 is greater than thewidth W_(S) (see FIG. 2 ) of each of the connecting portions, and alsothe length of the first base portion of the base portion 133 is greaterthan the width W_(s) of each of the connecting portions.

As above-described, each of the vibrational arms has the width W and thewidth W_(e) greater than the width W in this embodiment, but, thisinvention is not limited to this, namely, each of the vibrational armsmay comprise a plurality of different thicknesses having a firstthickness T and a second thickness T_(e) greater than the firstthickness T. In other words, each of the vibrational arms comprises afirst vibrational portion having the first thickness T and a secondvibrational portion having the second thickness T_(e) greater than thefirst thickness T. Namely, the width W is replaced with the thickness Tand the width W_(e) is replaced with the thickness T_(e). This inventionmay include the relationships of the thickness T_(e) greater than thethickness T and the width W_(e) greater than the width W. As is shown inFIG. 10 , at least one groove is formed in each of upper and lower facesof the first vibrational portion of each of the vibrating arms. Inaddition, the second vibrational portion having the width W_(e) and/orthe thickness T_(e) has opposite main surfaces and opposite sidesurfaces, e.g. the opposite main surfaces has a third main surface and afourth main surface and the opposite side surfaces has a third sidesurface and a fourth side surface, and e.g. a plurality of metal filmscomprising a first metal film having a first thickness and a secondmetal film having a second thickness greater than the first thicknessare disposed on at least one or each of the third and fourth mainsurfaces of the second vibrational portion of each of the vibrationalarms, and the first thickness of the first metal film is less than ahalf of the second thickness of the second metal film, preferably, onethird of the second thickness of the second metal film. Also, at leastone of the first and second metal films disposed on at least one of thethird and fourth main surfaces of the second vibrational portion of eachof the vibrational arms extends on at least one or each of the third andfourth side surfaces of the second vibrational portion of thecorresponding one of the vibrational arms and at least one of the firstand second metal films comprises gold or silver. For another instance,the second vibrational portion of each of the vibrational arms has athird main surface and a fourth main surface opposite the third mainsurface and each of the third and fourth main surfaces has a first mainportion and a second main portion. In addition, a first metal film isdisposed on each of the first and second main portions of at least oneof the third and fourth main surfaces of the second vibrational portionof each of the vibrational arms and a second metal film is disposed onthe first metal film disposed on each of the first and second mainportions of the at least one of the third and fourth main surfaces ofthe second vibrational portion of each of the vibrational arms. Also, athickness of the second metal film on the first metal film disposed onthe second main portion of the at least one of the third and fourth mainsurfaces of the second vibrational portion of each of the vibrationalarms is greater than a thickness of the second metal film on the firstmetal film disposed on the first main portion of the at least one of thethird and fourth main surfaces of the second vibrational portion of thecorresponding one of the vibrational arms, and the second metal film isdisposed on the first metal film disposed on the second main portion ofthe at least one of the third and fourth main surfaces of the secondvibrational portion of each of the vibrational arms so that the quartzcrystal tuning fork resonator has an oscillation frequency lower than32.75 kHz. In addition, a thickness of the first metal film disposed onthe first main portion of the at least one of the third and fourth mainsurfaces of the second vibrational portion of each of the vibrationalarms is substantially equal to a thickness of the first metal filmdisposed on the second main portion of the at least one of the third andfourth main surfaces of the second vibrational portion of thecorresponding one of the vibrational arms and the first metal filmcomprises chromium or nickel. For further instance, the secondvibrational portion of each of the vibrational arms has a third sidesurface and a fourth side surface opposite the third side surface, and afifth side surface free in vibration. The third side surface isconnected to the fourth side surface through the fifth side surface, anda third electrode is disposed on the third side surface, a fourthelectrode is disposed on the fourth side surface and a fifth electrodeis disposed on the fifth side surface, the third electrode disposed onthe third side surface is connected to the fourth electrode disposed onthe fourth side surface through the fifth electrode disposed on thefifth side surface to prevent the resonator from chipping end portionsof the vibrational arms thereof when shocked.

Similar to this, a quartz crystal tuning fork resonator 140 of a seventhembodiment of the present invention is shown in FIG. 11 showing a planview thereof. The resonator 140 comprises vibrational arms 141 and 142and a base portion 143 attached to the vibrational arms, and the baseportion 143 has a mounting arm 144 protruding from the base portion. Inthis embodiment, the mounting arm 144 is between the vibrational arms141 and 142 and the vibrational arms 141 and 142 have grooves 145 and146, respectively. For this case the width W is also less than the widthW_(e) similar to that of FIG. 10 .

Especially, the quartz crystal tuning fork resonator of the fourthembodiment to the seventh embodiment can be miniaturized with a smallseries resistance R₁ and a high quality factor Q₁ because it has a widthW_(e) greater than a width W, and the width W_(e) operates as a mass.

Next, a value of a piezoelectric constant e′₁₂ is described, which is ofgreat importance and necessary to excite a quartz crystal tuning forkresonator capable of vibrating in a flexural mode of the presentinvention. The larger a value of the piezoelectric constant e′₁₂becomes, the higher electromechanical transformation efficiency becomes.The piezoelectric constant e′₁₂ of the present invention can be definedby a function of the cutting angles θ_(y), θ_(x) and θ_(z) shown in FIG.1 , and piezoelectric constants e₁₁=0.171 C/m² and e₁₄=−0.0406 C/m² ofquartz crystal. In order to obtain a quartz crystal tuning forkresonator, capable of vibrating in a flexural mode and having a smallseries resistance R₁ and a high quality factor Q, the piezoelectricconstant e′₁₂ of the present invention is within a range of 0.1 C/m² to0.19 C/m² in the absolute value. It is, therefore, easily understoodthat this value is enough large to obtain the quartz crystal tuning forkresonator with a small series resistance R₁ and a high quality factor Q.

Especially, in order to obtain a quartz crystal tuning fork resonatorcapable of vibrating in a flexural mode with a much smaller seriesresistance R₁, the piezoelectric constant e′₁₂ is preferably within arange of 0.12 C/m² to 0.19 C/m² in the absolute value.

In addition, as an example, a quartz crystal tuning fork resonatorcomprises a plurality of vibrational arms having a first vibrational armand a second vibrational arm, and a groove having a first steppedportion and a second stepped portion is formed in at least one of afirst main surface and a second main surface of each of the first andsecond vibrational arms, in which a first electrode is disposed on thefirst stepped portion of the groove, a second electrode is disposed onthe second stepped portion of the groove, and a third electrode isdisposed on each of side surfaces of each of the first and secondvibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) isbetween the first electrode and the third electrode disposed opposite tothe first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) isbetween the second electrode and the third electrode disposed oppositeto the second electrode, and in which the piezoelectric constantse′_(12i) and e′_(12o) are within the range of 0.12 C/m² to 0.19 C/m² inthe absolute value, respectively, and a product of the e′_(12i) and thee′_(12o) is greater than 0.

As an another example, a quartz crystal tuning fork resonator comprisesa plurality of vibrational arms having a first vibrational arm and asecond vibrational arm, and a through hole having a first side surfaceand a second side surface is formed in each of the first and secondvibrational arms, in which a first electrode is disposed on the firstside surface of the through hole, a second electrode is disposed on thesecond side surface of the through hole, and a third electrode isdisposed on each of side surfaces of each of the first and secondvibrational arms, in which the piezoelectric constant e′₁₂(=e′_(12i)) isbetween the first electrode and the third electrode disposed opposite tothe first electrode, and the piezoelectric constant e′₁₂(=e′_(12o)) isbetween the second electrode and the third electrode disposed oppositeto the second electrode, and in which the piezoelectric constantse′_(12i) and e_(12o) are within the range of 0.12 C/m² to 0.19 C/m² inthe absolute value, respectively, and a product of the e′_(12i) and thee′_(12o) is greater than 0.

Therefore, the quartz crystal tuning fork resonator described above hasa small series resistance R₁ and a high quality factor Q, and also afrequency of high stability.

FIG. 12 shows a plan view of a width-extensional mode quartz crystalresonator 150 constructing an electronic apparatus of the presentinvention, and which vibrates in a width-extensional mode. Thewidth-extensional mode resonator 150 comprises a vibrational portion151, connecting portions 152, 152 a and supporting portions 153 and 154connected to a mounting portion 155 constructing the supportingportions. Namely, the vibrational portion 151 is connected to thesupporting portions 153, 154 having the mounting portion 155 through theconnecting portions 152, 152 a. In addition, an electrode 151 a isdisposed on an obverse surface of the vibrational portion 151 and anelectrode 151 b (not visible) is disposed on a reverse surface of thevibrational portion 151.

In more detail, the electrode 151 a disposed on the obverse surface ofthe vibrational portion 151 is connected to an electrode 153 a disposedon the mounting portion 155, while the electrode 151 b disposed on thereverse surface of the vibrational portion 151 is connected to anelectrode 154 a disposed on the mounting portion 155 through anelectrode 154 b disposed on a side surface of the mounting portion.Namely, a pair of electrodes is disposed on the obverse and reversesurfaces of the vibrational portion 151. Also, the vibrational portion151 has a width W₀ and a length L₀. In general, a ratio W₀/L₀ is lessthan 0.35. In addition, the mounting portion 155 is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In thisembodiment, a cutting angle of the resonator is within a range of ZYlwt(80° to 100°)/(−10° to)+10°/(75° to +115°) and a piezoelectric constante′₃₁ of the resonator is within a range of 0.1 C/m² to 0.19 C/m² in theabsolute value to obtain the resonator with a small series resistance R₁and a high quality factor Q.

Similar to this, a length-extensional mode quartz crystal resonator canbe obtained by replacing the width W₀ with the length L₀. In this casethe connecting portions are formed in the width direction. In thisembodiment, a cutting angle of the length-extensional mode resonator iswithin a range of ZYlwt (80° to 100°)/(−10° to)+10°/(−35° to +35°) and apiezoelectric constant e′₃₂ of the resonator is within a range of 0.12C/m² to 0.19 C/m² in the absolute value to obtain the resonator with asmall series resistance R₁ and a high quality factor Q.

FIG. 13(a) and FIG. 13(b) show a plan view of a thickness shear modequartz crystal resonator 156 constructing an electronic apparatus of thepresent invention and a F-F′ sectional view of the thickness shear moderesonator 156 capable of vibrating in a thickness shear mode. Theresonator 156 comprises a vibrational portion 157 having a width W₀, alength L₀ and a thickness T₀, and electrodes 158 and 159 are disposed onan obverse surface and a reverse surface so that the electrodes haveopposite electrical polarity each other. Namely, a pair of electrodes isdisposed on the vibrational portion 157. Also, the resonator 156 ishoused in a package comprising a case for housing the resonator and alid for covering an open end of the case. In this embodiment, a cuttingangle of the resonator is within a range of ZYlwt (−5° to)+5°/±(37° to58°)/(85° to 95°) and a piezoelectric constant e′34 of the resonator iswithin a range of 0.055 C/m² to 0.14 C/m² in the absolute value toobtain the resonator with a small series resistance R₁ and a highquality factor Q. In order to obtain much smaller series resistance R₁,the piezoelectric constant e′₃₄ of the resonator is preferably within arange of 0.085 C/m² to 0.12 C/m² in the absolute value.

FIG. 14 shows a plan view of a Lame mode quartz crystal resonator 210constructing an electronic apparatus of the present invention, andvibrating in a Lame mode. The Lame mode resonator 210 comprises avibrational portion 211, connecting portions 212, 213 and supportingportions 214, 215 having mounting portions 216 and 217. Namely, thevibrational portion 211 is connected to the supporting portions 214, 215through the connecting portions 212, 213. In addition, an electrode 218is disposed on an obverse surface of the vibrational portion 211 and anelectrode 219 (not visible) is disposed on a reverse surface of thevibrational portion 211.

In more detail, the electrode 218 disposed on the obverse surface of thevibrational portion 211 is extended into the mounting portion 217, whilethe electrode 219 disposed on the reverse face of the vibrationalportion 211 is extended into the mounting portion 216. Namely, a pair ofelectrodes is disposed on the obverse and reverse surfaces of thevibrational portion 151. Also, the vibrational portion 211 has a widthW₀ and a length L₀. In general, a ratio L₀/W₀ is approximately equal tom for a fundamental mode of vibration and an overtone mode of vibrationof the resonator, where m is an order of vibration of the resonator andan integer. For example, when a Lame mode quartz crystal resonator hasone of m=1, 2, 3 and n, the resonator vibrates in a fundamental mode form=1, a second overtone mode for m=2, a third overtone mode for m=3 andan n^(th) overtone mode for m=n.

Also, the m has a close relationship to the number of electrodesdisposed on the vibrational portion. For example, when an electrode isdisposed opposite each other on both of an obverse surface and a reversesurface of the vibrational portion, this is called “the number of twoelectrodes”, in other words, “a pair of electrodes”. Namely, thevibrational portion has the number of p electrodes, where p is an evennumber such as 2, 4, 6, 8 and 10. As an example, when p of thevibrational portion comprises 6, the vibrational portion vibrates in athird overtone mode. In this example, three pairs of electrodes aredisposed on the vibrational portion. Namely, the third overtone mode ofvibration is a principal vibration. As is apparent from thisrelationship, there is a relationship of m=p/2.

Therefore, the principal vibration vibrates in the order of vibrationcorresponding to the number of an electrode pair or electrode pairs. Forexample, the principal vibration vibrates in a fundamental mode ofvibration, a second overtone mode of vibration and a third overtone modeof vibration, respectively, for an electrode pair, two electrode pairsand three electrode pair disposed on the vibrational portion. In detail,when m_(e) electrode pairs are disposed on the vibrational portion, theprincipal vibration vibrates in an n^(th) overtone mode of vibration,and m_(e) corresponds to the n, where m_(e) is an integer. It isneedless to say that this concept can be applied to a width-extensionalmode quartz crystal resonator and a length-extensional mode quartzcrystal resonator.

In more detail, an even number of electrodes are disposed on the obverseand reverse surfaces of the vibrational portion and the electrodesdisposed opposite each other has an opposite electrical polarity. Inaddition, each of the mounting portions 216, 217 is mounted on amounting portion of a package comprising a case for housing theresonator and a lid for covering an open end of the case. In thisembodiment, a cutting angle of the resonator is within a range of ZYlwt(−5° to)+5°/±(35° to 60°)/±(40° to) 50° and a piezoelectric constante′32 of the resonator is within a range of 0.045 C/m² to 0.13 C/m² inthe absolute value to obtain the resonator with a small seriesresistance R₁ and a high quality factor Q.

FIG. 15(a) and FIG. 15(b) show a plan view of a resonator for sensingangular velocity constructing an electronic apparatus of the presentinvention and a G-G′ sectional view of the resonator. In thisembodiment, the resonator 220 comprises a quartz crystal tuning forkresonator, capable of vibrating in a flexural mode and comprisingvibrational arms 221, 222 and a base portion 223 attached to thevibrational arms, the base portion 223 is mounted on a mounting portionof a package comprising a case for housing the resonator and a lid forcovering an open end of the case. In addition, each of the vibrationalarms 221 and 222 has a first main surface and a second main surfaceopposite the first main surface and side surfaces, and the vibrationalarm 221 has grooves 224, 230, while the vibrational arm 222 has grooves225, 237, each of which has stepped portions comprising a first steppedportion and a second stepped portion. Also, a cutting angle of theresonator is within a range of ZYlwt (−20° to)+20°/(−25° to)+25°/(−18°to +18°) and a piezoelectric constant e′₁₂ of the resonator is within arange of 0.1 C/m² to 0.19 C/m² in the absolute value. The resonator ofthis embodiment can vibrate in a flexural mode of a fundamental mode andan inverse phase.

In more detail, electrodes 226 and 227 which are of the same electricalpolarity are disposed on the side surfaces such that an electrodeterminal H is defined, while an electrode 228 is disposed inside thegroove 224 and an electrode 229 which is of the same electrical polarityto the electrode 228 is disposed inside the groove 230 such that anelectrode terminal H′ of opposite electrical polarity to the electrodeterminal H is defined. Namely, two electrode terminals H-H′ for an inputsignal are constructed. On the other hand, electrodes 231, 232, 233which are of the same electrical polarity are disposed on the sidesurfaces and inside the grooves 225, 237 such that an electrode terminalI is defined, while electrodes 234, 235, 236 which are of the sameelectrical polarity are disposed on the side surfaces and inside thegrooves 225, 237 such that an electrode terminal I′ of oppositeelectrical polarity to the electrode terminal I is defined. Namely, twoelectrode terminals I-I′ for an output signal are constructed. Theresonator of this embodiment is made of quartz crystal, but may be madeof a piezoelectric material such as lithium tantalite, lithium niobateand ceramics.

FIG. 16(a) and FIG. 16(b) show a plan view and a J-J′ sectional view ofa quartz crystal resonator 230 of an eighth embodiment of the presentinvention, and which is a quartz crystal tuning fork resonator. Theresonator 230 comprises vibrational arms 231 and 232 and a base portion233 attached to the vibrational arms.

Similar to FIG. 15(a), though the base portion 233 has a constant width,this invention is not limited to this, but may include a shape of thebase portion which has a first base portion including a first length anda first width, and a second base portion including a second length and asecond width greater than the first width, each of the vibrational arms231 and 232 is connected to the first base portion of the base portion233 and the second base portion of the base portion 233 is mounted on amounting portion of a case and a lid is connected to the case to coveran open end of the case. In addition, when a ratio of the first lengthof the first base portion and the second length of the second baseportion is defined by a first ratio, the first ratio is within a rangeof 0.85 to 1.2.

Needless to say, this concept can be applied to a variety of resonators.To get a resonator with high shock-proof properties, e.g., the firstwidth of the first base portion is within a range of 0.17 mm to 0.38 mm,the first length of the first base portion is within a range of 0.1 mmto 0.23 mm, the second width of the second base portion is within arange of 0.32 mm to 0.5 mm, preferably within a range of 0.32 mm to 0.43mm, and the second length of the second base portion is within a rangeof 0.1 mm to 0.25 mm, preferably, within a range of 0.11 mm to 0.23 mm.

In addition, each of the vibrational arms 231 and 232 has a first mainsurface and a second main surface opposite the first main surface andside surfaces, and the first and second main surfaces have centrallinear portions 242, 243, respectively. The vibrational arm 231 hasgrooves 234, 235 and the vibrational arm 232 has grooves 235, 237. Inaddition, the grooves 234 and 236 are formed between the central linearportion 242 and an outer edge of the vibrational arm 231, respectively.Similar to this, the grooves 235 and 237 are formed between the centrallinear portion 243 and an outer edge of the vibrational arm 232,respectively

Moreover, each of the grooves 234, 235, 236 and 237 has a width W₈, anda width W₇ including a portion of the central linear lines 242 and 243is formed in each of the vibrational arms 231 and 232. In addition, adistance W₉ is formed in the width direction of the vibrational arms 231and 232 between an outer edge of the groove and an outer edge of thevibrational arms. In detail, a width W of the arms 231 and 232 hasgenerally a relationship of W=W₇+2W₈+2W₉, and the width W₈ isconstructed so that W₈≥W₇, W₉.

In this embodiment, also, the grooves are constructed at the arms sothat a ratio W₈/(W/2) of the width W₈ and a half width of the width W isgreater than 0.35 and less than 1, preferably, within a range of 0.35 to0.95. In addition, the width W₇ is less than 0.05 mm, preferably, lessthan 0.03 mm and as shown in FIG. 2 , the width W₈ which correspondswith the width W₂, is less than 0.07 mm, preferably, within a range of0.01 mm to 0.12 mm or within a range of 0.02 mm to 0.12 mm, morepreferably, within a range of 0.012 mm to 0.025 mm or within a range of0.02 mm to 0.04 mm, still more preferably, within a range of 0.02 mm to0.035 mm or within a range of 0.038 mm to 0.05 mm or within a range of0.015 mm to 0.05 mm to obtain a very large moment of inertia of thevibrational arms.

That is, the quartz crystal tuning fork resonator, capable of vibratingin a fundamental mode, and having a frequency of high stability can beprovided with a small series resistance a high quality factor Q₁ and asmall capacitance ratio r₁ because it has a very large electromechanicaltransformation efficiency. Needless less to say, the distance W₉corresponds with each of the distance W₁ and the distance W₃ shown inFIG. 2 because a groove is divided into two grooves having a firstgroove and a second groove by a central linear portion.

In FIG. 16(b), the vibrational arm 231 has grooves 234, 236, 238 and 240cut into it. The grooves 234, 236, 238 and 240 have a first set ofelectrodes 256, 257, 258 and 259 of the same electrical polarity, whilethe side surfaces of the arm 231 have a second set of electrodes 244 and245 having an opposite electrical polarity to the first set ofelectrodes 256, 257, 258 and 259. The vibrational arm 232 has grooves235, 237, 239 and 241 constructed in a similar manner as the vibrationalarm 231. The grooves 235, 237, 239 and 241 have a third set ofelectrodes 252, 253, 254 and 255 of the same electrical polarity, andthe side surfaces of the vibrational arm 232 have a fourth set ofelectrodes 250 and 251 with the opposite electrical polarity to thethird electrodes 252, 253, 254 and 255. The electrodes disposed on thevibrational arms 231 and 232 are connected as shown in FIG. 16(b),namely, two electrode terminals of different electrical polarity K-K′are obtained.

In detail, the first set of electrodes 256, 257, 258 and 259 disposed onthe grooves 234, 236, 238 and 240 of the vibrational arm 231 have thesame electrical polarity as the fourth set of electrodes 250 and 251disposed on both side surfaces of the vibrational arm 232, while thesecond set of electrodes 244 and 245 disposed on both side surfaces ofthe vibrational arm 231 have the same electrical polarity as the thirdset of electrodes 252, 253, 254 and 255 disposed on the grooves 235,237, 239 and 241 of the arm 232. When a direct current voltage isapplied between the electrode terminals K-K′, an electric field Exoccurs along the arrow direction inside the vibrational arms 231 and232. As the electric field Ex occurs perpendicular to the electrodesdisposed on the vibrational arms, as shown in the arrow signs, theelectric field Ex has a very large value and a large distortion occursat the vibrational arms. As a result, a quartz crystal tuning forkresonator capable of vibrating in a flexural mode is obtained with asmall series resistance R₁ and a high quality factor Q because even whenminiaturized there is a very large electromechanical transformationefficiency for the resonator.

FIG. 17 shows a plan view of a quartz crystal unit of a first embodimentof the present invention and omitting a lid. The quartz crystal unit 160comprises the quartz crystal tuning fork resonator 10 shown in FIG. 2 ,a case 165 for housing the resonator and a lid for covering an open endof the case (not shown here). Also, the resonator 10 has mounting arms36 and 37, each of which is mounted on a mounting portion 166 and amounting portion 167 of the case 165. In detail, an electrode disposedon the mounting arm 36 is connected to an electrode disposed on themounting portion 166 by adhesives 168 or a metal such as solder, andsimilarly, an electrode disposed on the mounting arm 37 is connected toan electrode disposed on the mounting portion 167 by adhesives 169 or ametal.

FIG. 18 shows a plan view of a quartz crystal unit of a secondembodiment of the present invention and omitting a lid. The quartzcrystal unit 170 comprises the quartz crystal tuning fork resonator 50shown in FIG. 5 , a case 175 for housing the resonator and a lid forcovering an open end of the case (not shown here). Namely, the resonatorcomprises vibrational arms and a base portion. Also, the case 175 hasmounting portions 176 and 177 and the resonator 50 comprises a mountingarm 77 having a first mounting portion with a first width and a secondmounting portion with a second width greater than the first width andprotruding from the base portion, which is mounted on the mountingportion 177 of the case 175. Namely, the second mounting portion withthe second width is mounted on the mounting portion 177 of the case 175.In detail, an electrode disposed on the mounting arm 77 is connected toan electrode disposed on the mounting portion 177 by adhesives 179 or ametal such as solder, and similarly, an electrode disposed on the baseportion of the resonator 50 is connected to an electrode disposed on themounting portion 176 by adhesives 178 or a metal such as solder.

FIG. 19 shows a cross-sectional view of a quartz crystal unit of a thirdembodiment of the present invention. The quartz crystal unit 180comprises a contour mode quartz crystal resonator 185 or a thicknessshear mode quartz crystal resonator 185, a case 181 and a lid 182. Inmore detail, the resonator 185 is mounted on a mounting portion 184 ofthe case 181 by conductive adhesives 76 or solder. Also, the case 181and the lid 182 are connected through a connecting member 183. Forexample, the contour mode resonator 185 in this embodiment is the sameresonator as one of the quartz crystal tuning fork resonators 10, 50,90, 110, 120, 130, 140, 220 and 230 described in detail in FIG. 2 -FIG.11 and FIG. 15 -FIG. 16 .

In this embodiment, circuit elements are connected at outside of thequartz crystal unit to get a quartz crystal oscillator. Namely, only thequartz crystal tuning fork resonator is housed in the unit and also, itis housed in the unit in vacuum. In this embodiment, the quartz crystalunit of a surface mounting type is shown, but the quartz crystal tuningfork resonator may be housed in a tubular type, namely, a quartz crystalunit of the tubular type. In addition, the quartz crystal unit of thepresent invention includes any shape of a quartz crystal unit comprisinga quartz crystal resonator, a case and a lid to house the quartz crystalresonator in vacuum. As an example of any shape of the quartz crystalunit, when the quartz crystal tuning fork resonator 10 is formed in aquartz crystal wafer, an end portion of the mounting arm 36 is notconnected to an end portion of the mounting arm 37, as is shown in FIGS.2 and 17 , but the end portion of the mounting arm 36 may be connectedto the end portion of the mounting arm 36 in the quartz crystal wafer toget a connected (closed) mounting arm. In detail, the connected (closed)mounting arm comprises one end portion and the other end portion eachconnected to the base portion. Also, each of the connected mounting armand the base portion has an upper face and a lower face, namely, a firstsurface and a second surface opposite the first surface. For this case,a quartz crystal unit comprises a case and a lid, and each of the caseand the lid has an interior space and an open end. Also, the lower face(the second surface) of each of the connected mounting arm and the baseportion is mounted on a mounting portion of the case and the upper face(the first surface) of each of the connected mounting arm and the baseportion is mounted on a mounting portion of the lid, namely, the lowerface (the second surface) of each of the connected mounting arm and thebase portion is connected to the open end of the case and the upper face(the first surface) of each of the connected mounting arm and the baseportion is connected to the open end of the lid to cover the open end ofeach of the case and the lid. A width of the open end of each of the lidand the case is less, preferably, equal to, more preferably, greaterthan a width of the connected mounting arm and/or the base portion toget a big connected power. When each of the case and the lid has nothrough hole, at least one of the open end of the case and open end ofthe lid is connected to the corresponding one of the upper and lowerfaces of each of the connected mounting arm and the base portion so thatthe quartz crystal tuning fork resonator is maintained in a vacuum, andwhen one of the case and the lid has a through hole including a firstdiameter and a second diameter greater than the first diameter, a metalor a glass is disposed into the through hole of the second diameter toclose the through hole of one of the case and the lid in a vacuum afterthe open end of each of the case and the lid is connected to thecorresponding one of the upper and lower faces of each of the connectedmounting arm and the base portion. As above-described, the base portionis located between the open end of the case and the open end of the lid,for example, a part having an area of the base portion is locatedbetween the open end of the case and the open end of the lid. It isneedless to say that the quartz crystal tuning fork tines are locatedbetween the interior space of the case and the interior space of thelid. Also, the connection of the open end of the case is performedsimultaneously with the connection of the open end of the lid, but,according to the present invention, the connection of the open end ofthe case may be performed in a step different from the connection of theopen end of the lid, namely, the connection of the open end of the caseis performed after or before the connection of the open end of the lidis performed. Also, a first electrode (metal film) is disposed on eachof a surface of the open end of the lid and the upper face of each ofthe connected mounting arm and the base portion and a second electrode(metal film) is disposed on each of a surface of the open end of thecase and the lower face of each of the connected mounting arm and thebase portion. The lid is connected to the connected mounting arm and thebase portion through the first electrode disposed on the surface of theopen end of the lid and the first electrode disposed on the upper faceof each of the connected mounting arm and the base portion, while thecase is connected to the connected mounting arm and the base portionthrough the second electrode disposed on the surface of the open end ofthe case and the second electrode disposed on the lower face of each ofthe connected mounting arm and the base portion. Namely, each of theconnection of the lid and the connected mounting arm and the baseportion and the connection of the case and the connected mounting armand the base portion is performed by an anode connection method. Inaddition, the first electrode disposed on each of the surface of theopen end of the lid and the upper face of each of the connected mountingarm and the base portion has an electrical polarity opposite to anelectrical polarity of the second electrode disposed on each of thesurface of the open end of the case and the lower face of each of theconnected mounting arm and the base portion. Also, the case and the lidare made of a piezoelectric material such as quartz crystal or a glassor ceramics and have a thermal expansion coefficient less than that ofthe quartz crystal tuning fork resonator. In addition, the presentinvention includes the following example, namely, one of the case andthe lid comprises a plurality of through holes having a first throughhole and a second through hole and an electrode is disposed on a surfaceof each of the first and second through holes. In addition, one of thecase and the lid comprises a first electrode and a second electrode eachof which is disposed on an outer surface of the one of the case and thelid. Also, the vibrational arms shown in FIG. 3 having two electrodeterminals including first and second electrode terminals are locatedbetween an inner surface in the interior space of the case and an innersurface in the interior space of the lid, and the first electrodeterminal of the vibrational arms is connected to the first electrodedisposed on the outer surface of the one of the case and the lid throughthe electrode disposed on the surface of the first through hole and thesecond electrode terminal of the vibrational arms is connected to thesecond electrode disposed on the outer surface of the one of the caseand the lid through the electrode disposed on the surface of the secondthrough hole. In this embodiment, one of the case and the lid comprisesthe first and second through holes, but this invention is not limited tothis, namely, the lid comprises an outer surface on which a firstelectrode is disposed and a first through hole having a surface on whichan electrode is disposed and the case comprises an outer surface onwhich a second electrode is disposed and a second through hole having asurface on which an electrode is disposed. For this case, the firstelectrode terminal of the vibrational arms is connected to the firstelectrode disposed on the outer surface of the lid through the electrodedisposed on the surface of the first through hole of the lid and thesecond electrode terminal of the vibrational arms is connected to thesecond electrode disposed on the outer surface of the case through theelectrode disposed on the surface of the second through hole of thecase. In addition, a metal or a glass is disposed into each of the firstand second through holes to close each of the first and second throughholes and at least one of the first and second through holes is closedin a vacuum using the metal or the glass.

Also, instead of the quartz crystal tuning fork resonator and thethickness shear mode quartz crystal resonator, one of alength-extensional mode quartz crystal resonator, a width-extensionalmode quartz crystal resonator and a Lame mode quartz crystal resonatorwhich are a contour mode quartz crystal resonator, respectively, or aSAW (Surface Acoustic Wave) resonator or a piezoelectric resonator forsensing angular velocity (angular velocity sensor) made of quartzcrystal or ceramics may be housed in the unit.

In addition, a member of the case and the lid is ceramics or glass and ametal or glass, respectively, and a connecting member is a metal orglass with low melting point. Also, a relationship of the resonator, thecase and the lid described in this embodiment is applied to a quartzcrystal oscillator of the present invention which will be described inFIG. 20 .

FIG. 20 shows a cross-sectional view of a quartz crystal oscillator of afirst embodiment of the present invention. The quartz crystal oscillator190 comprises a quartz crystal oscillating circuit, a case 191 and a lid192. In this embodiment, circuit elements constructing the oscillatingcircuit are housed in a quartz crystal unit comprising a contour modequartz crystal resonator 195 or a thickness shear mode quartz crystalresonator 195, the case 191 and the lid 192. Also, the oscillatingcircuit of this embodiment comprises an amplifier 197 including afeedback resistor, the resonator 195, a plurality of capacitors (notshown here) and a drain resistor (not shown here), and a CMOS inverteris used as the amplifier 197.

In addition, in this embodiment, the resonator 195 is mounted on amounting portion 194 of the case 191 by conductive adhesives 196 orsolder. As described above, the amplifier 197 is housed in the quartzcrystal unit and mounted on the case 191. Also, the case 191 and the lid192 are connected through a connecting member 193.

FIG. 21 shows a diagram of an embodiment of a quartz crystal oscillatingcircuit constructing a quartz crystal oscillator of the presentinvention. The quartz crystal oscillating circuit 201 comprises anamplifier (CMOS inverter) 202, a feedback resistor 204, a drain resistor207, a plurality of capacitors 205, 206 and a quartz crystal resonator203. Namely, the oscillating circuit 201 comprises an amplificationcircuit 208 having the amplifier 202 and the feedback resistor 204, anda feedback circuit 209 having the drain resistor 207, the capacitors205, 206 and the quartz crystal resonator 203. The quartz crystalresonator 203 is one of the resonators already described above. Forexample, when the resonator 203 is a quartz crystal tuning forkresonator capable of vibrating in a flexural mode, an output signal ofthe oscillating circuit 201 is outputted through a buffer circuit (notshown in FIG. 21 ), and is an oscillating frequency of a fundamentalmode of vibration of the resonator.

In other words, the oscillation frequency of the fundamental mode ofvibration of the quartz crystal tuning fork resonator is outputted fromthe oscillating circuit through the buffer circuit as an output signal.According to the present invention, a nominal frequency of thefundamental mode of vibration of the quartz crystal tuning forkresonator is within a range of 10 kHz to 200 kHz. Especially, afrequency of 32.768 kHz is very available for use in an electronicapparatus of the present invention. In general, the output signal has anoscillation frequency which is within a range of −100 PPM to +100 PPM tothe nominal frequency, e.g. 32.768 kHz.

In more detail, the quartz crystal oscillator in this example comprisesa quartz crystal oscillating circuit and a buffer circuit, namely, thequartz crystal oscillating circuit comprises an amplification circuitand a feedback circuit, and the amplification circuit comprises anamplifier (CMOS inverter) and a feedback resistor and the feedbackcircuit comprises a quartz crystal tuning fork resonator capable ofvibrating in a flexural mode, a drain resistor and a plurality ofcapacitors. Also, the quartz crystal tuning fork resonator alreadydescribed in FIG. 2 -FIG. 11 and FIG. 15 -FIG. 16 is used in a quartzcrystal oscillator of the present invention. Instead of the quartzcrystal tuning fork resonator, an another contour mode quartz crystalresonator such as a length-extensional mode quartz crystal resonator, awidth-extensional mode quartz crystal resonator and a Lame mode quartzcrystal resonator or a thickness shear mode quartz crystal resonator ora resonator for sensing angular velocity may be used.

FIG. 22 shows a diagram of the feedback circuit of FIG. 21 . In thisembodiment, the feedback circuit has a quartz crystal tuning forkresonator capable of vibrating in a flexural mode. Now, when angularfrequency co, of the quartz crystal tuning fork resonator 203, aresistance value R_(d) of the drain resistor 207, capacitance valuesC_(g), C_(d) of the capacitors 205, 206, crystal impedance R_(ei) of thequartz crystal resonator 203, an input voltage V₁, and an output voltageV₂ are taken, a feedback rate β_(i) is defined by β_(i)=|V₂|i/|V₁|_(i),where i shows a vibration order, for example, when i=1 and 2, β₁ and β₂are a feedback rate for a fundamental mode of vibration and a secondovertone mode of vibration of the resonator, respectively.

In addition, load capacitance C_(L) is given by C_(L)=C_(g)C_(d)/(C_(g)+C_(d)), and when C_(g)=C_(d)=C_(g)d and R_(d)>>R_(ei), thefeedback rate β_(i) is given by β_(i)=1/(1+kC_(L) ²), where k isexpressed by a function of ω_(i), R_(d) and R_(ei). Also, R_(ei) isapproximately equal to series resistance R_(i) of the resonator.

Thus, it is easily understood from a relationship of the feedback rateβ_(i) and load capacitance C_(L) that the feedback rate of a resonancefrequency for the fundamental mode of vibration and the overtone modevibration becomes large, respectively, according to decrease of loadcapacitance C_(L). Therefore, when C_(L) has a small value, anoscillation of the overtone mode occurs very easily, instead of that ofthe fundamental mode. This is the reason why maximum amplitude of theovertone mode of vibration becomes smaller than that of the fundamentalmode of vibration, and the oscillation of the overtone mode satisfies anamplitude condition and a phase condition simultaneously which are thecontinuous condition of an oscillation in an oscillating circuit.

As it is also one object of the present invention to provide a quartzcrystal oscillator having a flexural mode, quartz crystal tuning forkresonator, capable of vibrating in a fundamental mode and having afrequency of high stability (high time accuracy) of an output signal,and having reduced electric current consumption, load capacitance C_(L)is less than 25 pF in this embodiment to reduce electric currentconsumption. To get much reduced electric current consumption, C_(L) ispreferably less than 15 pF because the electric current consumption isproportional to C_(L). More preferably, C_(L) is greater than 2 pF andless than 15 pF to satisfy each of the reduced electric currentconsumption and a phase condition of a second overtone mode of vibrationof the resonator insufficient in an oscillation circuit of theoscillator.

In addition, in order to suppress the second overtone mode of vibrationof the resonator and to obtain the quartz crystal oscillator having anoutput signal of an oscillation frequency of a fundamental mode ofvibration of the resonator, the quartz crystal oscillator comprising anamplification circuit and a feedback circuit is constructed so that itsatisfies a relationship of α₁/α₂>β₂/β₁ and where α₁ and α₂ are,respectively, an amplification rate of the fundamental mode of vibrationand the second overtone mode of vibration of the amplification circuit,and β₁ and β₂ are, respectively, a feedback rate of the fundamental modeof vibration and the second overtone mode of vibration of the feedbackcircuit.

In other words, the quartz crystal oscillator is constructed so that aratio of the amplification rate α₁ of the fundamental mode of vibrationand the amplification rate α₂ of the second overtone mode of vibrationof the amplification circuit is greater than that of the feedback rateβ₂ of the second overtone mode of vibration and the feedback rate Pi ofthe fundamental mode vibration of the feedback circuit, and also aproduct of the amplification rate α₁ and the feedback rate β₁ of thefundamental mode of vibration is greater than 1. A description of afrequency of high stability in the quartz crystal oscillator will beperformed later.

Also, characteristics of the amplifier of the amplification circuitconstructing the quartz crystal oscillating circuit of this embodimentcan be expressed by negative resistance −RL_(i). For example, when i=1,negative resistance −RL₁ is for a fundamental mode of vibration of theresonator and when i=2, negative resistance −RL₂ is for a secondovertone mode of vibration of the resonator. In this embodiment, thequartz crystal oscillating circuit is constructed so that a ratio of anabsolute value of negative resistance −RL₁ of the fundamental mode ofvibration of the amplification circuit and series resistance R₁ of thefundamental mode of vibration is greater than that of an absolute valueof negative resistance −RL₂ of the second overtone mode of vibration ofthe amplification circuit and series resistance R₂ of the secondovertone mode of vibration.

That is, the oscillating circuit is constructed so that it satisfies arelationship of |−RL₁|/R₁>|−RL₂|/R₂. By constructing the oscillatingcircuit like this, an oscillation of the second overtone mode can besuppressed, as a result of which a frequency of oscillation of thefundamental mode of vibration can be output as an output signal becausethe oscillation of the fundamental mode generates easily in theoscillating circuit. In more detail, an absolute value of the negativeresistance −RL₁ is greater than 55 kΩ and less than 800 kΩ, preferably,within a range of 60 kΩ to 500 kΩ, more preferably, within a range of 60kΩ to 285 kΩ, and an absolute value of the negative resistance −RL₂ isless than 200 kΩ, preferably, less than 105 kΩ, more preferably, lessthan 80 kΩ to suppress the frequency of oscillation of the secondovertone mode of the quartz crystal tuning fork resonator in theoscillating circuit and to obtain the frequency of oscillation of thefundamental mode thereof.

In this embodiment, a quartz crystal tuning fork resonator is used, but,instead of the tuning fork resonator, an another contour mode quartzcrystal resonator such as a width-extensional mode quartz crystalresonator, a length-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator may be used in a quartz crystal oscillatorof the present invention. In this case, a principal vibration of thecontour mode quartz crystal resonator is outputted from an oscillatingcircuit constructing the quartz crystal oscillator through a buffercircuit. In order to suppress a sub-vibration of the contour mode quartzcrystal resonator, the quartz crystal oscillator comprising anamplification circuit and a feedback circuit is constructed so that itsatisfies a relationship of α_(p)/α_(s)>β_(s)/β_(p) and α_(p)β_(p)>1,where α_(p) and α_(s) are, respectively, an amplification rate of theprincipal vibration and the sub-vibration of the amplification circuit,and β_(p) and β_(s) are, respectively, a feedback rate of the principalvibration and the sub-vibration of the feedback circuit.

Similar to the quartz crystal tuning fork resonator, for the contourmode quartz crystal resonator, a quartz crystal oscillating circuit ofthe present invention is constructed so that a ratio of an absolutevalue of negative resistance −RL_(p) of the principal vibration of theamplification circuit and a series resistance R_(p) of the principalvibration is greater than that of an absolute value of negativeresistance −RL_(s) of the sub-vibration of the amplification circuit anda series resistance R_(s) of the sub-vibration. That is, the oscillatingcircuit is constructed so that it satisfies a relationship of|−RL_(p)|/R_(p)>|−RL_(s)|/R_(s). By constructing the oscillating circuitlike this, an oscillation of the sub-vibration can be suppressed, as aresult of which a frequency of oscillation of the principal vibrationcan be outputted as an output signal because the oscillation of theprincipal vibration generates easily in the oscillating circuit. Inaddition, the principal vibration and the sub-vibration have the samemode of vibration and a different order of vibration each other.

FIG. 23 shows a block diagram of an embodiment of an electronicapparatus of the present invention, and illustrating the diagram of afacsimile apparatus. As shown in FIG. 23 , the apparatus generallycomprises a modem, a phonetic circuit, a timepiece circuit, a printingportion, a taking portion, an operation portion and a display portion.In this principle, perception and scanning of reflection light of lightprojected on manuscript in the taking portion are performed by CCD(Charge Coupled Device), in addition, light and shade of the reflectionlight are transformed into a digital signal, and the signal is modulatedby the modem and is sent to a phone line (communication line). Also, ina receiving side, a received signal is demodulated by the modem and isprinted on a paper in the print portion by synchronizing the receivedsignal with a signal of a sending side. In addition, the display portioncomprises at least one of a liquid crystal display (LCD) portion, aplasma display panel (PDP) portion, a surface-conductionelectron-emitter display (SED) portion and an organicelectroluminescence display (OED) portion.

In FIG. 23 , a quartz crystal resonator is used as a CPU clock of thecontrol portion and the printing portion, as a clock of the phoneticcircuit and the modem, and as a time standard of the timepiece. Namely,the resonator constructs a quartz crystal oscillator and an outputsignal of the oscillator is used. Like this, a plurality of oscillatorsis used for the electronic apparatus. For example, it is used as asignal to display time at the display portion. In this case, a quartzcrystal tuning fork resonator, capable of vibrating in a flexural modeis generally used, and e.g. as the CPU clock, a contour mode quartzcrystal resonator such as a length-extensional mode quartz crystalresonator, a width-extensional mode quartz crystal resonator and a Lamemode quartz crystal resonator or a thickness shear mode quartz crystalresonator is used. To get the facsimile apparatus of this embodimentwhich operates normally, an accuracy signal of output of the oscillatoris required for the facsimile apparatus, which is one of the electronicapparatus of the present invention. Also, a digital display and ananalogue display are included in the display of the present invention.

In this embodiment, though the facsimile apparatus is shown as anexample of an electronic apparatus of the present invention, the presentinvention is not limited to this, namely, the present invention includesall electronic apparatus, each of which comprises a quartz crystaloscillator and a display portion at least, for example, cellar phones,telephones, a TV set, cameras, a video set, video cameras, pagers,personal computers, printers, CD players, MD players, electronic musicalinstruments, car navigators, car electronics, timepieces, IC cards andso forth. In addition, the electronic apparatus may have an anotheroscillator comprising a piezoelectric resonator for sensing angularvelocity made of quartz crystal, ceramics, lithium tantalite and lithiumniobate. Also, the electronic apparatus of the present invention maycomprise a battery (cell), e.g. a lithium battery or a fuel cell whichis housed in the electronic apparatus of the present invention.

Thus, the electronic apparatus of the present invention comprising adisplay portion and a quartz crystal oscillator at least may operatenormally because the quartz crystal oscillator comprises the quartzcrystal oscillating circuit with a frequency of high stability, namely,a frequency of high reliability.

Moreover, capacitance ratios r₁ and r₂ of a flexural mode, quartzcrystal tuning fork resonator are given by r₁=C₀/C₁ and r₂=C₀/C₂,respectively, where C₀ is shunt capacitance in an electrical equivalentcircuit of the resonator, and C₁ and C₂ are, respectively, motionalcapacitance of a fundamental mode of vibration and a second overtonemode of vibration in the electrical equivalent circuit of the resonator.In addition, the resonator has a quality factor Q₁ for the fundamentalmode of vibration and a quality factor Q₂ for the second overtone modeof vibration. Also, the motional capacitance C₁ of the fundamental modeof vibration of the resonator is greater than the motional capacitanceC₂ of the second overtone mode of vibration thereof from therelationship of r_(i) less than r₂ as above-described. In addition, aratio (L₁/L₂) of a motional inductance L₁ of the fundamental mode ofvibration of the resonator and a motional inductance L₂ of the secondovertone mode of vibration thereof is less than 42 approximately fromthe relationships of r₁ less than r₂ and ω₂=6.5ω₁ approximately, whereω₁ and ω₂ represent an angular frequency of the fundamental mode ofvibration and the second overtone mode of vibration, respectively, ofthe resonator. Also the ratio (L₁/L₂) is less than 6.5(Q₁/Q₂) from therelationship of R₁<R₂. As a result, an output signal having anoscillation frequency of the fundamental mode of vibration of theresonator is output in an oscillating circuit through a buffer circuitbecause a phase condition of the fundamental mode of vibration is muchbetter than that of the second overtone mode of vibration in theoscillating circuit.

In detail, the tuning fork resonator of this embodiment is provided sothat the influence on resonance frequency of the fundamental mode ofvibration by the shunt capacitance becomes smaller than that of thesecond overtone mode of vibration by the shunt capacitance, namely, sothat it satisfies a relationship of S₁=r₁/2Q₁ ²<S₂=r₂/2Q₂ ², preferably,S₁<S₂/2, where S₁ and S₂ are called “a stable factor of frequency” ofthe fundamental mode of vibration and the second overtone mode ofvibration. As a result, the tuning fork resonator, capable of vibratingin the fundamental mode and having a frequency of high stability can beprovided because the influence on the resonance frequency of thefundamental mode of vibration by the shunt capacitance C₀ is asextremely small as it can be ignored. In this embodiment S₂ has a valuegreater than 0.13×10⁻⁶ to suppress the second overtone mode of vibrationof the resonator.

In addition, as described above, it will be easily understood that thequartz crystal resonator comprising vibrational arms and a base portion,according to the present invention, may have outstanding effects.Similar to this, the quartz crystal unit and the quartz crystaloscillator, according to the present invention, may have alsooutstanding effects. In addition, the electronic apparatus comprisingthe quartz crystal oscillator comprising the quartz crystal oscillatingcircuit having the quartz crystal tuning fork resonator, capable ofvibrating in a flexural mode, and having novel shapes, novel electrodeconstruction and excellent electrical characteristics, according to thepresent invention, may have the outstanding effects. Similar to this, itwill be easily understood that the electronic apparatus comprising thequartz crystal oscillator comprising the quartz crystal oscillatingcircuit having the another contour mode quartz crystal resonator or thethickness shear mode quartz crystal resonator or the resonator forsensing angular velocity, according to the present invention, may havealso the outstanding effect. In addition to this, while the presentinvention has been shown and described with reference to preferredembodiments thereof, it will be understood by those skilled in the artthat the changes in shape and electrode construction can be made thereinwithout departing from the spirit and scope of the present invention.

What is claimed is:
 1. A quartz crystal unit comprising: a case; a lid;and a quartz crystal resonator having a base portion, and first andsecond tuning fork arms connected to the base portion, the base portionhaving a length less than 0.5 mm and greater than a spaced-apartdistance between the first tuning fork arm and the second tuning forkarm, each of the first and second tuning fork arms having a width lessthan 0.1 mm and a length less than 1.56 mm, and a plurality of differentwidths including a first width and a second width greater than the firstwidth, and opposite main surfaces, at least one groove being formed inat least one of the opposite main surfaces of each of the first andsecond tuning fork arms so that a length of the at least one grooveformed in the at least one of the opposite main surfaces of each of thefirst and second tuning fork arms is within a range of 0.3 mm to 0.79mm, the quartz crystal resonator being housed in the case, and the lidbeing connected to the case.
 2. The quartz crystal unit according toclaim 1; wherein each of the first and second tuning fork arms comprisesa first vibrational portion having the first width, and a secondvibrational portion having the second width greater than the first widthso that the first vibrational portion is formed between the secondvibrational portion and the base portion, the first vibrational portionof each of the first and second tuning fork arms having the oppositemain surfaces including a first main surface and a second main surface;wherein the at least one groove comprises a groove formed in at leastone of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms so that a widthof the groove formed in the at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms is less than 0.07 mm, a first distance in thewidth direction of the groove measured from a first outer edge of thegroove to a first outer edge of the corresponding one of the first andsecond tuning fork arms being defined by W₁ and a second distance in thewidth direction of the groove measured from a second outer edge oppositethe first outer edge of the groove to a second outer edge opposite thefirst outer edge of the corresponding one of the first and second tuningfork arms being defined by W₃, and so that each of the first distance W₁and the second distance W₃ is less than 0.015 mm; wherein an overalllength of the quartz crystal resonator is less than 2.1 mm and a widthof the base portion is less than 0.55 mm; wherein the case has amounting portion; wherein the base portion is mounted on the mountingportion of the case; and wherein the lid is connected to the case tocover an open end of the case.
 3. The quartz crystal unit according toclaim 2; wherein the overall length of the quartz crystal resonator iswithin a range of 1.02 mm to 1.95 mm; wherein the length of the baseportion is within a range of 0.015 mm to 0.49 mm and the width of thebase portion is within a range of 0.15 mm to 0.53 mm; wherein the lengthof each of the first and second tuning fork arms is within a range of0.56 mm to 1.15 mm; wherein the first vibrational portion of each of thefirst and second tuning fork arms has a first length and the secondvibrational portion of each of the first and second tuning fork arms hasa second length; wherein the first length of the first vibrationalportion of each of the first and second tuning fork arms is within arange of 0.32 mm to 0.72 mm and greater than the second length of thesecond vibrational portion of the corresponding one of the first andsecond timing fork arms; wherein the second width of the secondvibrational portion of each of to first and second tuning fork arms isless than the second length of the second vibrational portion of thecorresponding one of the first and second tuning fork arms; wherein thebase portion has a first base portion, a second base portion and atleast one cut portion formed between the first and second base portions,each of the first and second tuning fork arms being connected to thefirst base portion of the base portion; and wherein the second baseportion of the base portion is mounted on the mounting portion of thecase.
 4. The quartz crystal unit according to claim 2; wherein the firstwidth of the first vibrational portion of each of the first and secondtuning fork arms is defined by W; wherein a spaced-apart distancebetween the first vibrational portion of the first tuning fork arm andthe first vibrational portion of the second tuning fork arm is definedby W₄; wherein the sum of the width W of the first vibrational portionof each of the first and second tuning fork arms and the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning fork armis defined by a width W₅=2W+W₄; wherein the width W₅ is within a rangeof 0.15 mm to 0.52 mm; wherein the overall length of the quartz crystalresonator is within a range of 0.8 mm to 1.95 mm and the length of thebase portion is within a range of 0.015 mm to 0.49 mm; wherein the baseportion has a first base portion, a second base portion and at least onecut portion formed between the first and second base portions, each ofthe first and second tuning fork arms being connected to the first baseportion of the base portion; and wherein the second base portion of thebase portion is mounted on the mounting portion of the case.
 5. Thequartz crystal unit according to claim 2; wherein the first vibrationalportion of each of the first and second tuning fork arms has a firstlength within a range of 0.32 mm to 0.72 mm, and the second vibrationalportion of each of the first and second tuning fork arms has a secondlength within a range of 0.17 mm to 0.38 mm; wherein the first length ofthe first vibrational portion of each of the first and second tuningfork arms is greater than the second length of the second vibrationalportion of the corresponding one of the first and second tuning forkarms; wherein the base portion has a first base portion including afirst length, a second base portion and at least one cut portion formedbetween the first and second base portions, each of first and secondtuning fork arms being connected to the first base portion of the baseportion; wherein the second base portion of the base portion is mountedon the mounting portion of the case; wherein the second length of thesecond vibrational portion of each of the first and second tuning forkarms is greater than the first length of the first base portion of thebase portion; and wherein the overall length of the quartz crystalresonator is within a range of 0.8 mm to 1.95 mm and the length of thebase portion is within a range of 0.015 mm to 0.49 mm.
 6. The quartzcrystal unit according to claim 2; wherein the first width of the firstvibrational portion of each of the first and second tuning fork arms isdefined by W; wherein a spaced-apart distance between the firstvibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by W₄;wherein the sum of the width W of the first vibrational portion of eachof the first and second tuning fork arms and the spaced-apart distanceW₄ between the first vibrational portion of the first tuning fork armand the first vibrational portion of the second tuning fork arm isdefined by a width W₅=2W+W₄; wherein the width W₅ is within a range of0.15 mm to 0.52 mm; wherein the length of the base portion is within arange of 0.015 mm to 0.49 mm; wherein the second vibrational portion ofeach of the first and second tuning fork arms has a second length; andwherein the second width of the second vibrational portion of each ofthe first and second tuning fork arms is less than the second length ofthe second vibrational portion of the corresponding one of the first andsecond tuning fork arms and greater than a spaced-apart distance betweenthe second vibrational portion of the first tuning fork arm and thesecond vibrational portion of the second tuning fork arm.
 7. The quartzcrystal unit according to claim 2; wherein the overall length of thequartz crystal resonator is within a range of 0.8 mm to 1.95 mm and thelength of the base portion is within a range of 0.015 mm to 0.49 mm;wherein the first vibrational portion of each of the first and secondtuning fork arms has a first length and the first width defined by W;wherein the second vibrational portion of each of the first and secondtuning fork arms has a second length and the second width defined by We;wherein the first length of the first vibrational portion of each of thefirst and second tuning fork arms is greater than the second length ofthe second vibrational portion of the corresponding one of the first andsecond tuning fork arms; wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms isless than a first spaced-apart distance between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm and greater than a second spaced-apartdistance between the second vibrational portion of the first tuning forkarm and the second vibrational portion of the second tuning fork arm;wherein a ratio of the second width We of the second vibrational portionof each of the first and second tuning fork arms and the first width Wof the first vibrational portion of the corresponding one of the firstand second tuning fork arms is defined by Wr=We/W; wherein the ratio Wris within a range of 2.1 to 2.9; wherein the base portion has a firstbase portion, a second base portion and at least one cut portion formedbetween the first and second base portions, each of the first and secondtuning fork arms being connected to the first base portion of the baseportion; and wherein the second base portion of the base portion ismounted on the mounting portion of the case.
 8. The quartz crystal unitaccording to claim 2; wherein the length of the base portion is within arange of 0.015 mm to 0.49 mm; wherein the base portion has a first baseportion including a first length, a second base portion including asecond length less than the first length, and at least one cut portionformed between the first and second base portions, each of the first andsecond tuning fork arms being connected to the first base portion of thebase portion; wherein the second base portion of the base portion ismounted on the mounting portion of the case; wherein the secondvibrational portion of each of the first and second tuning fork arms hasa second length; and wherein the second length of the second vibrationalportion of each of the first and second tuning fork arms is greater thanthe first length of the first base portion of the base portion.
 9. Thequartz crystal unit according to claim 2; wherein the length of each ofthe first and second tuning fork arms is within a range of 0.53 mm to1.46 mm, the overall length of the quartz crystal resonator being withina range of 1.02 mm to 1.95 mm and the length of the base portion beingwithin a range of 0.015 mm to 0.49 mm; wherein the width of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is within a range of 0.015 mm to 0.05 mm and defined by W₂ and thefirst width of the first vibrational portion of each of the first andsecond tuning fork arms is defined by W; wherein a ratio of the width W₂of the groove formed in the at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms and the first width W of the first vibrationalportion of the corresponding one of the first and second tuning forkarms is defined by W₂₁=W₂/W; and wherein the ratio W₂₁ is within a rangeof 0.35 to 0.95.
 10. The quartz crystal unit according to claim 1;wherein the opposite main surfaces of each of the first and secondtuning fork arms have a first main surface and a second main surface,each of the first and second mam surfaces of earn of the first andsecond tuning fork arms having a central linear portion; wherein the atleast one groove comprises a groove formed in the central linear portionof at least one of the first and second main surfaces of each of thefirst and second tuning fork arms so that a width of the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of each of the first and second tuning fork arms isless than 0.07 mm, a first distance in the width direction of the groovemeasured from a first outer edge of the groove to a first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that each of the first distance W₁ and the second distanceis less than 0.015 mm; wherein an overall length of the quartz crystalresonator is less than 2.1 mm; wherein a width of the base portion isless than 0.55 mm and the length of the base portion is within a rangeof 0.015 mm to 0.49 mm; wherein at least one mounting arm has a lengthwithin a range of 0.3 mm to 1.85 mm and is connected to the base portionthrough a connecting portion having a width W_(S) less than 0.41 mm sothat the at least one mounting arm extends in a common direction withthe first and second tuning fork arms outside the first and secondtuning fork arms; wherein the case has a mounting portion; wherein theat least one mounting arm is mounted on the mounting portion of thecase; and wherein the lid is connected to the case to cover an open endof the case.
 11. The quartz crystal unit according to claim 10; whereinthe width W_(S) of the connecting portion is within a range of 0.015 mmto 0.14 mm; wherein the length of the base portion is defined by L₁ anda length of the connecting portion is defined by L₂; wherein the lengthL₁ of the base portion is greater than the length L₂ of the connectingportion; wherein the at least one mounting arm has a first arm portionincluding a first width and a second arm portion including a secondwidth greater than the first width; wherein the second arm portion ofthe at least one mounting arm is mounted on the mounting portion of thecase; wherein a first electrode is disposed on a surface of the grooveformed in the central linear portion of the at least one of the firstand second main surfaces of each of the first and second tuning forkarms; wherein a second electrode is disposed on a surface of theconnecting portion; wherein a third electrode is disposed on a surfaceof the at least one mounting arm; wherein a fourth electrode is disposedon a surface of the mounting portion of the case; and wherein the firstelectrode disposed on the surface of the groove formed in the centrallinear portion of the at least one of the first and second main surfacesof one of the first and second tuning fork arms is connected to thethird electrode disposed on the surface of the at least one mounting armthrough the second electrode disposed on the surface of the connectingportion, and the third electrode disposed on the surface of the at leastone mounting arm is connected to the fourth electrode disposed on thesurface of the mounting portion of the case.
 12. The quartz crystal unitaccording to claim 10; wherein the overall length of the quartz crystalresonator is within a range of 1.02 mm to 1.95 mm; wherein the width ofeach of the first and second tuning fork arms is greater than 0.05 mmand less than 0.1 mm, and the length of each of the first, and secondtuning fork arms is within a range of 0.53 mm to 1.46 mm; wherein thebase portion has a first base portion, a second base portion and two cutportions formed between the first and second base portions, each of thefirst and second tuning fork arms being connected to the first baseportion of the base portion; wherein the at least one mounting arm hasfirst and second mounting arms each including a length within a range of0.3 mm to 1.85 mm; wherein the first mounting arm is connected to thesecond base portion of the base portion through a first connectingportion and the second mounting arm is connected to the second baseportion of the base portion through a second connecting portion so thatthe second base portion of the base portion and the first and secondconnecting portions have a concave shape or a U-shape with the first andsecond mounting arms, a width of each of the first and second mountingarms being less than 0.45 mm, and a width W_(S) of each of the first andsecond connecting portions being less than 0.41 mm.
 13. The quartzcrystal unit according to claim 12; wherein the length of the baseportion is defined L₁ and a length of the connecting portion is definedby L₂; wherein the length L₁ of the base portion is greater than thelength L₂ of the connecting portion; wherein the length L₂ of theconnecting portion is within a range of 0.04 mm to 0.5 mm and greaterthan the width W_(S) of the connecting portion; wherein each of thefirst and second mounting arms has a plurality of arm portions includingfirst, second, third, fourth and fifth arm portions, each of the first,third and fifth arm portions of each of the first and second mountingarms extending substantially parallel to the first and second tuningfork arms, the first arm portion being connected to the third armportion through the second arm portion extending not parallel to thefirst and second tuning fork arms, the third arm portion being connectedto the fifth arm portion through the fourth arm portion extending notparallel to the first and second tuning fork arms, a direction of thesecond arm portion being different from that of the fourth arm portion;wherein the case has first and second mounting portions; wherein thefifth arm portion of the first mounting arm is mounted on the firstmounting portion of the case; wherein the fifth arm portion of thesecond mounting arm is mounted on the second mounting portion of thecase; and wherein the first arm portion of the first mounting arm isconnected to the second base portion of the base portion through thefirst connecting portion and the first arm portion of the secondmounting arm is connected to the second base portion of the base portionthrough the second connecting portion so that the second base portion ofthe base portion and the first and second connecting portions have aU-shape or a concave shape with the first arm portions of the first andsecond mounting arms, the width of each of the first and second mountingarms being within a range of 0.025 mm to 0.4 mm, and the width W_(S) ofeach of the first and second connecting portions being within a range of0.015 mm to 0.14 mm.
 14. The quartz crystal unit according to claim 12;wherein each of the first and second tuning fork arms comprises at leasttwo vibrational portions having a first vibrational portion including agenerally tapered shape comprised of a plurality of different widthshaving the first width and the second width greater than the first widthand a first length, and a second vibrational portion including a thirdwidth less than or equal to the second width and a second length lessthan the first length, the first vibrational portion of each of thefirst and second tuning fork arms having the opposite main surfacesincluding a third main surface and a fourth main surface, each of thethird and fourth main surfaces of the first vibrational portion of eachof the first and second tuning fork arms having a central linearportion; and wherein the at least one groove comprises a groove formedin the central linear portion of at least one of the third and fourthmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms so that a length of the groove formed in thecentral linear portion of the at least one of the third and fourth mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms is within a range of 0.72 mm to 0.785 mm, and awidth of the groove formed in the central linear portion of the at leastone of the third and fourth main surfaces of the first vibrationalportion of each of the first and second tuning fork arms is less than0.07 mm and greater than each of a first distance in the width directionof the groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms and a second distance in the width direction of the groove measuredfrom a second outer edge opposite the first outer edge of the groove toa second outer edge opposite the first outer edge of the correspondingone of the first and second tuning fork arms, each of the first distanceand the second distance being less than 0.015 mm, the first width of thefirst vibrational portion of each of the first and second tuning forkarms being greater than 0.05 mm and less than 0.057 mm, and the secondwidth of the first vibrational portion of each of the first and secondtuning fork arms being greater than 0.058 mm and less than 0.1 mm. 15.The quartz crystal unit according to claim 12; wherein the width of eachof the first and second mounting arms is within a range of 0.025 mm to0.4 mm, and the width W_(S) of each of the first and second connectingportions is within a range of 0.015 mm to 0.14 mm; and wherein the widthof each of the first and second mounting arms is greater than the widthof each of the first and second tuning fork arms.
 16. The quartz crystalunit according to claim 12; wherein the first base portion of the baseportion has a length; wherein each of the first and second connectingportions has a length; and wherein the length of each of the first andsecond connecting portions is greater than the length of the first baseportion of the base portion.
 17. The quartz crystal unit according toclaim 1; wherein the opposite main surfaces of each of the first andsecond tuning fork arms have a first main surface and a second mainsurface, each of the first and second main surfaces of each of the firstand second tuning fork arms having a central linear portion; wherein theat least one groove comprises a groove formed in the central linearportion of at least one of the first and second main surfaces of each ofthe first and second tailing fork arms so that a width of the grooveformed in the central linear portion of the at least one of the firstand second main surfaces of each of the first and second tuning forkarms is less than 0.07 mm, a first distance in the width direction ofthe groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms being defined by W₁ and a second distance in the width direction ofthe groove measured from a second outer edge opposite the first outeredge of the groove to a second outer edge opposite the first outer edgeof the corresponding one of the first and second tuning fork arms beingdefined by W₃, and so that each of the first distance W₁ and the seconddistance W₃ is less than 0.015 mm; wherein an overall length of thequartz crystal resonator is within a range of 1.02 mm to 1.95 mm;wherein a width of the base portion is less than 0.55 mm; wherein atleast one mounting arm is connected to the base portion through aconnecting portion and has a length within a range of 0.3 mm to 1.85 mmso that the at least one mounting arm extends in a common direction withthe first and second tuning fork arms outside the first and secondtuning fork arms; wherein the at least one mounting arm has a first armportion including a first width and a second arm portion including asecond width greater than the first width; wherein the case has amounting portion; wherein the second arm portion of the at least onemounting arm is mounted on the mounting portion of the case; wherein anelectrode is disposed on each of a surface of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms, a surface ofthe connecting portion, a surface of the at least one mounting arm and asurface of the mounting portion of the case; wherein the electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of one of the first and second tuningfork arms is connected to the electrode disposed on the surface of theat least one mounting arm through the electrode disposed on the surfaceof the connecting portion, and the electrode disposed on the surface ofthe at least one mounting arm is connected to the electrode disposed onthe surface of the mounting portion of the case; and wherein the lid isconnected to the case to cover an open end of the case.
 18. The quartzcrystal unit according to claim 17; wherein the width of the grooveformed in the central linear portion of the at least one of the firstand second main surfaces of each of the first and second tuning harkarms is within a range of 0.038 mm to 0.05 mm; wherein each of the firstdistance W₁ and the second distance W₃ is within a range of 0.001 mm0.012 mm; wherein the length of the base portion is defined by L₁ and alength of the connecting portion is defined by L₂; wherein the length L₁of the base portion is greater than the length L₂ of the connectingportion; wherein the first width of each of the first and second tuningfork arms is defined by W and a spaced-apart distance between the firsttuning fork arm and the second tuning fork atm is defined by W₄; whereinthe sum of the first width W of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first tuning fork armand the second tuning fork arm is defined a width W₅=2W+W₄; wherein thewidth W₅ is less than the width of the base portion; wherein theconnecting portion is connected to the at least one mounting arm and hasan outer edge, one of the first and second tuning fork arms adjoiningthe at least one mounting arm and having an outer edge; wherein adistance in the length direction of the connecting portion measured fromthe outer edge of the connecting portion to the Outer edge of one of thefirst and second tuning fork arms is defined by L₄; and wherein thedistance L₄ is within a range of 0.012 mm to 0.38 mm.
 19. The quartzcrystal unit according to claim 17; wherein the at least one mountingarm comprises a mounting arm having a plurality of arm portionsincluding first, second, third, fourth and fifth arm portions, each ofthe first, third and fifth arm portions of the mounting arm extendingsubstantially parallel to the first and second tuning fork arms, thefirst arm portion being connected to the third arm portion through thesecond arm portion extending not parallel to the first and second tuningfork arms, the third arm portion being connected to the fifth armportion through the fourth arm portion extending not parallel to thefirst and second tuning fork arms, a direction of the second arm portionbeing different from that of the fourth arm portion; wherein the firstarm portion of the mounting arm is connected to the base portion throughthe connecting portion; and wherein the fifth arm portion of themounting arm is mounted on the mounting portion of the case; wherein thelength of the base portion is within a range of 0.015 mm to 0.49 mm;wherein the base portion has a first base portion, a second base portionand a third base portion connecting the first base portion to the secondbase portion, two cut portions being formed between the first and secondbase portions, each of the first and second tuning fork arms beingconnected to the first base portion of the base portion; wherein thefirst arm portion of the mounting arm is connected to the second baseportion of the base portion through the connecting portion so that theconnecting portion has a L-shape with the first arm portion of themounting arm; wherein the mounting arm adjoins one of the first andsecond timing fork arms; wherein a spaced-apart distance between thethird base portion of the base portion and the first arm portion of themounting arm is defined by a first distance and a spaced-apart distancebetween one of the first and second tuning fork arms and the third armportion of the mounting arm is defined by a second distance; and whereinthe first distance is greater than the second distance.
 20. The quartzcrystal unit according to claim 17; wherein the length of the baseportion is within a range of 0.015 mm to 0.49 mm; wherein the at leastone mounting arm has first and second mounting arms each including alength within a range of 0.3 mm to 1.85 mm; wherein the first mountingarm is connected to the base portion through a first connecting portionand the second mounting arm is connected to the base portion through asecond connecting portion, each of the first and second, mounting armsextending in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms; wherein the case hasfirst and second mounting portions; the first mounting arm is mounted onthe first mounting portion of the case and the second mounting arm ismounted on the second mounting portion of the case; wherein each of thefirst and second connecting portions has a width W_(S) and a length L₂;and wherein the length L₂ of each of the first and second connectingportions is within a range of 0.04 mm to 0.5 mm and greater than thewidth W_(S) of each of the first and second connecting portions.
 21. Thequartz crystal unit according to claim 17; wherein the length of thebase portion is within a range of 0.015 mm to 0.49 mm; wherein the atleast one mounting arm has first and second mounting arms each includinga length within a range of 0.3 mm to 1.85 mm; wherein the first mountingarm is connected to the base portion through a first connecting portionand the second mounting arm is connected to the base portion through asecond connecting portion, each of the first and second mounting armsextending in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms; wherein the case hasfirst and second mounting portions; wherein the first mounting arm ismounted on the first mounting portion of the case and the secondmounting arm is mounted on the second mounting portion of the case;wherein the groove formed in the central linear portion of the at leastone of the first and second main surfaces of each of the first andsecond tuning fork arms is formed in the central linear portion of eachof the first and second main surfaces of each of the first and secondtuning fork arms; wherein a first electrode is disposed on a surface ofthe groove formed in the central linear portion of each of the first andsecond main surfaces of each of the first and second tuning fork arms;wherein a second electrode is disposed on a surface of each of the firstand second connecting portions; wherein a third electrode is disposed ona surface of each of the first and second mounting arms; wherein afourth electrode is disposed on a surface of each of the first andsecond mounting portions of the case; wherein the first electrodedisposed on the surface of the groove formed in the central linearportion of each of the first and second main surfaces of one of thefirst and second tuning fork arms is connected to the third electrodedisposed on the surface of the first mounting arm through the secondelectrode disposed on the surface of the first connecting portion, andthe third electrode disposed on the surface of the first mounting arm isconnected to the fourth electrode disposed on the surface of the firstmounting portion of the case; and wherein the first electrode disposedon the surface of the groove formed in the central linear portion ofeach of the first and second main surfaces of the other of the first andsecond tuning fork arms is connected to the third electrode disposed onthe surface of the second mounting arm through the second electrodedisposed on the surface of the second connecting portion, and the thirdelectrode disposed on the surface of the second mounting arm isconnected to the fourth electrode disposed on the surface of the secondmounting portion of the case.
 22. The quartz crystal unit according toclaim 1; wherein an overall length of the quartz crystal resonator iswithin a range of 1.02 mm to 1.95 mm; wherein the length of the baseportion is within a range of 0.015 mm to 0.49 mm; wherein each of thefirst and second tuning fork arms each having the first width and thesecond width greater than the first width has the opposite main surfacesincluding a first main surface and a second main surface, each of thefirst and second main surfaces of each of the first and second tuningfork arms having a central linear portion; wherein the at least onegroove comprises a groove formed in the central linear portion of atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms so that a width of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms is less than0.07 mm and a length of the groove formed in the central linear portionof the at least one of the first and second main surfaces of each of thefirst and second tuning fork arms is within a range of 0.72 mm to 0.785mm, a first distance in the width direction of the groove measured froma first outer edge of the groove to first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that each of the first distance W₁ and the second distanceW₃ is less than 0.015 mm; wherein the base portion has a first baseportion including a first width, a second base portion including asecond width equal to the first width, and a third base portionconnecting the first base portion to the second base portion, the thirdbase portion having a third length and a third width greater than thethird length and less than each of the first width of the first baseportion and the second width of the second base portion, the third widthof the third base portion being within a range of 0.045 mm to 0.18 mm,each of the first and second tuning fork arms being connected to thefirst base portion of the base portion; wherein a first mounting arm isconnected to the second base portion of the base portion through a firstconnecting portion and a second mounting arm is connected to the secondbase portion of the base portion through a second connecting portion sothat the second base portion of the base portion and the first andsecond connecting portions have a concave shape or a U-shape with thefirst and second mounting arms, a length of each of the first and secondmounting arm being within a range of 0.3 mm to 1.85 mm; wherein the casehas first and second mounting portions; wherein the first mounting armis mounted on the first mounting portion of the case and the secondmounting arm is mounted on the second mounting portion of the case; andwherein the lid is connected to the case to cover an open end of thecase.
 23. The quartz crystal unit according to claim 22; wherein each ofthe first and second connecting portions has a length L₂ and a widthW_(S); wherein the length L₂ of each of the first and second connectingportions is greater than or equal to the width W_(S) of each of thefirst and second connecting portions, the width W_(S) of each of thefirst and second connecting portions being within a range of 0.015 mm to0.14 mm and the length L₂ of each of the first and second connectingportions being within a range of 0.02 mm to 0.18 mm; wherein the firstbase portion of the base portion has a first length; wherein the lengthL₂ of each of the first and second connecting portions is greater thanthe first length of the first base portion of the base portion; whereinthe third length of the third base portion is within a range of 0.015 mmto 0.08 mm; wherein each of the first and second mounting arms has aplurality of arm portions including first, second and third armportions, each of the first and third arm portions of each of the firstand second mounting arms extending substantially parallel to the firstand second tuning fork arms, the first arm portion being connected tothe third arm portion through the second arm portion extending notparallel to the first and second tuning fork arms; wherein the first armportion of the first mounting arm is connected to the second baseportion of the base portion through the first connecting portion and thefirst arm portion of the second mounting arm is connected to the secondbase portion of the base portion through the second connecting portionso that the second base portion of the base portion and the first andsecond connecting portions have a U-shape or a concave shape with thefirst arm portions of the first and second mounting arms, the firstmounting arm adjoining the first tuning fork arm and the second mountingarm adjoining the second tuning fork arm; wherein a spaced-apartdistance between the third base portion of the base portion and thefirst arm portion of the first mounting arm is defined by a firstdistance and a spaced-apart distance between the first tuning fork armand the third arm portion of the first mounting arm is defined by asecond distance; wherein the first distance is greater than the seconddistance; wherein a spaced-apart distance between the third base portionof the base portion and the first arm portion of the second mounting armis defined by a third distance and a spaced-apart distance between thesecond tuning fork arm and the third arm portion of the second mountingarm is defined by a fourth distance; and wherein the third distance isgreater than the fourth distance.
 24. The quartz crystal unit accordingto claim 22; wherein a width of the base portion is less than 0.55 mm;wherein the length of the base portion is defined by L₁ and the lengthof each of the first and second connecting portions is defined by L₂;wherein the length L₁ of the base portion is greater than the length L₂of each of the first and second connecting portions; wherein adifference of the length L₁ and the length L₂ is defined by L₁₂=L₁−L₂;wherein the difference L₁₂ is within a range of 0.1 mm to 0.32 mm;wherein the width of each of the first and second tuning fork arms isdefined by W; wherein the spaced-apart distance between the first tuningfork arm and the second tuning fork arm is defined by W₄; wherein thesum of the width W of each of the first and second tuning fork arms andthe spaced-apart distance W₄ between the first tuning fork arm and thesecond tuning fork arm is defined by a width W₅=2W+W₄; wherein the widthW₅ is within a range of 0.15 mm to 0.52 mm and less than the width ofthe base portion; wherein the first mounting arm is connected to thefirst connecting portion and adjoins the first tuning fork arm, each ofthe first connecting portion and the first tuning fork arm having anouter edge; wherein a distance in the length direction of the firstconnecting portion measured from the outer edge of the first connectingportion to the outer edge of the first tuning fork arm is defined by L₄;and wherein the distance L₄ is within a range of 0.012 mm to 0.38 mm.25. The quartz crystal unit according to claim 22; wherein each of thefirst and second tuning fork arms comprises at least two vibrationalportions having a first vibrational portion including the first widthand the second width greater than the first width and a first length,and a second vibrational portion including a third width less than orequal to the second width and a second length less than the firstlength, the first vibrational portion of each of the first and secondtuning fork arms having the opposite main surfaces including a thirdmain surface and a fourth main surface, each of the third and fourthmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms having a central linear portion; and wherein theat least one groove comprises a groove formed in the central linearportion of at least one of the third and fourth main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms so that a width of the groove formed in the central linear portionof the at least one of the third and fourth main surfaces of the firstvibrational portion of each of the first and second tuning fork arms isless than 0.07 mm and a length of the groove formed in the centrallinear portion of the at least one of the third and fourth main surfacesof the first vibrational portion of each of the first and second tuningfork arms is within a range of 0.72 mm to 0.785 mm, a first distance inthe width direction of the groove measured from a first outer edge ofthe groove to a first outer edge of the corresponding one of the firstand second tuning fork arms being defined by W₁ and a second distance inthe width direction of the groove measured from a second outer edgeopposite the first outer edge of the groove to a second outer edgeopposite the first outer edge of the corresponding one of the first andsecond tuning fork arms being defined by W₃, and so that each of thefirst distance W₁ and the second distance W₃ is less than 0.015 mm andthe first distance W₁ is greater than or equal to the second distanceW₃, the first width of the first vibrational portion of each of thefirst and second tuning fork arms being greater than 0.05 mm and lessthan 0.057 mm, and the second width of the first vibrational portion ofeach of the first and second tuning fork arms being greater than 0.058mm and less than 0.1 mm.
 26. The quartz crystal unit according to claim1; wherein each of the first and second tuning fork arms comprises afirst vibrational portion having the first width, and a secondvibrational portion having the second width greater than the firstwidth, the first vibrational portion of each of the first and secondtuning fork arms having the opposite main surfaces including a firstmain surface and a second main surface, each of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms having a central linear portion; wherein the atleast one groove comprises a groove formed in the central linear portionof at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms sothat a width of the groove formed in the central linear portion of theat least one of the first and second main surfaces of the firstvibrational portion of each of trig first and second tuning fork arms isless than 0.07 mm, a first distance in the width direction of the groovemeasured from a first outer edge of the groove to a first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that each of the first distance W₁ and the second distanceW₃ is less than 0.015 mm; wherein an overall length of the quartzcrystal resonator is less than 2.1 mm; wherein the length of the baseportion is within a range of 0.01.5 rain to 0.49 mm and a width of thebase portion is less than 0.55 mm; wherein a first mounting arm isconnected to the base portion through a first connecting portion and asecond mounting arm is connected to the base portion through a secondconnecting portion, each of the first and second mounting arms extendingin a common direction with the first and second tuning fork arms outsidethe first and second tuning fork arms, and having a length within arange of 0.3 mm to 1.85 mm; wherein the case has first and secondmounting portions; wherein the first mounting arm is mourned on thefirst mounting portion of the case and the second mounting arm ismourned on the second mounting portion of the case; and wherein the lidis connected to the case to cover an open end of the case.
 27. Thequartz crystal unit according to claim 26; wherein the base portion hasa first base portion including a first length and a first width, asecond base portion including a second length and a second width equalto the first width, and a third base portion connecting the first baseportion to the second base portion and including a third length and athird width less than each of the first and second widths, two cutportions being formed between the first and second base portions, thefirst length, of the first base portion being greater than each of thesecond length of the second base portion and the third length of thethird base portion, each of the first and second tuning fork arms beingconnected to the first base portion of the base portion; wherein thefirst mounting arm is connected to the second base portion of the baseportion through the first connecting portion and the second mounting armis connected to the second base portion of the base portion through thesecond connecting portion so that the second base portion of the baseportion and the first and second connecting portions have a concaveshape or a U-shape with the first and second mounting arms; wherein theoverall length of the quartz crystal resonator is within a range of 1.02mm to 1.95 mm and the width of the base portion is within a range of0.15 mm to 0.53 mm; wherein a width of each of the first and secondmounting arms is defined by W₆ and a width of each of the first andsecond connecting portions is defined by W_(S); wherein the width W₆ ofeach of the first and second mounting arms is less than 0.45 mm and thewidth W_(S) of each of the first and second connecting portions is lessthan 0.41 mm; wherein the length of the base portion is defined b L₁ alength of each of the first and second connecting portions is defined byL₂; wherein the length L₁ of the base portion is greater than the lengthL₂ of each of the first and second connecting portions; wherein adifference of the length L₁ and the length L₂ is defined by L₁₂=L₁−L₂;and wherein the difference L₁₂ is within a range of 0.1 mm to 0.32 mm.28. The quartz crystal unit according to claim 26; wherein the width ofthe base portion is within a range of 0.21 mm to 0.38 mm; wherein thefirst mounting arm is connected to the first connecting portion andadjoins the first tuning fork arm, each of the first connecting portionand the first tuning fork arm having an outer edge; wherein a distancein the length direction of the first connecting portion measured fromthe outer edge of the first connecting portion to the outer edge of thefirst tuning fork arm is defined by L₄; wherein the distance L₄ iswithin a range of 0.012 mm to 0.38 mm; wherein the width W₆ of each ofthe first and second mounting arms is within a range of 0.025 mm to 0.4mm and the width W_(S) of each of the first and second connectingportions is within a range of 0.015 mm to 0.14 mm; wherein the secondwidth of the second vibrational portion of each of the first and secondtuning fork arms is defined by We and the first width of the firstvibrational portion of each of the first and second tuning fork arms isdefined by W; wherein a ratio of the second width We of the secondvibrational portion of each of the first and second tuning fork arms andthe first width W of the first vibrational portion of the correspondingone of the first and second tuning fork arms is defined by W₂₁=We/W; andwherein the ratio W₂₁ is within a range of 1.8 to 3.6.
 29. The quartzcrystal unit according to claim 26; wherein the first vibrationalportion of each of the first and second tuning fork arms has a firstlength and the second vibrational portion of each of the first andsecond tuning fork arms has a second length; wherein each of the firstand second tuning fork arms comprises a third vibrational portion formedbetween the first and second vibrational portions and having a thirdlength and a third width greater than the first width of the firstvibrational portion and less than the second width of the secondvibrational portion; wherein the third length of the third vibrationalportion of each of the first and second tuning fork arms is less thaneach of the first length of the first vibrational portion of thecorresponding one of the first and second tuning fork arms and thesecond length of the second vibrational portion of the corresponding oneof the first and second tuning fork arms; wherein the third width of thethird vibrational portion of each of the first and second tuning forkarms is defined by Wm and the first width of the first vibrationalportion of each of the first and second tuning fork arms is defined byW; wherein a ratio of the third width Wm of the third vibrationalportion of each of the first and second tuning fork arms and the firstwidth W of the first vibrational portion of the corresponding one of thefirst and second tuning fork arms is defined by Wmo=Wm/W; and whereinthe ratio Wino is within a range of 1.6 to 2.3.
 30. The quartz crystalunit according to claim 26; wherein the first width of the firstvibrational portion of each of the first and second tuning fork arms isdefined by W; wherein the width W₆ of each of the first and secondmounting arms is less than or equal to the first width W of the firstvibrational portion of each of the first and second tuning fork arms;wherein a length of the groove formed in the central linear portion ofthe at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms isdefined by Lo and the length of each of the first and second tuning forkarms is defined by L; wherein a ratio of the length Lo the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms and the length L of the correspondingone of the first and second tuning fork arms is defined by Lga=Lo/L; andwherein the ratio Lga is within a range of 0.472 to 0.758.
 31. Thequartz crystal unit according to claim 26; wherein each of the firstdistance W₁ and the second distance W₃ is within a range of 0.0041 mm to0.0062 mm; wherein the overall length of the quartz crystal resonator iswithin a range of 1.02 mm to 1.95 mm: wherein the first vibrationalportion of each of the first and second tuning fork arms has a firstlength within a range of 0.32 mm to 0.72 mm; wherein the secondvibrational portion of each of the first and second tuning fork arms hasa second length within a range of 0.17 mm to 0.38 mm and less than afirst length of the first vibrational portion of the corresponding oneof the first and second tuning fork arms; wherein the second vibrationalportion of each of the first and second tuning fork arms has a firstportion including a third width and a third length, and a second portionincluding a fourth width greater than the third width and a fourthlength greater than the third length; wherein the first portion of thesecond vibrational portion of the first tuning fork arm is formedbetween the second portion of the second vibrational portion of thefirst tuning fork arm and the first vibrational portion of the firsttuning fork arm; wherein the first portion of the second vibrationalportion of the second tuning fork arm is formed between the secondportion of the second vibrational portion of the second tuning fork armand the first vibrational portion of the second tuning fork arm; whereinthe base portion has a first base portion including a first width, asecond base portion including a second width equal to the first width,and a third base portion connecting the first base portion to the secondbase portion and including a third width less than each of the first andsecond widths, two cut portions being formed between the first andsecond base portions, each of the first and second tuning fork armsbeing connected to the first base portion of the base portion; whereinthe first mounting arm is connected to the second base portion of thebase portion through the first connecting portion and the secondmounting arm is connected to the second base portion of the base portionthrough the second connecting portion so that the second base portion ofthe base portion and the first and second connecting portions have aconcave shape or a U-shape with the first and second mounting arms, eachof the first and second mounting arms having a width W₆; wherein thewidth W₆ of each of the first and second mounting arms is less than orequal to the first width of the first vibrational portion of each of thefirst and second tuning fork arms, each of the first and secondconnecting portion having a width W_(S) and a length L₂; wherein thewidth W_(S) of the first connecting portion is less than the length L₂of the first connecting portion and the width W_(S) of the secondconnecting portion is less than the length L₂ of the second connectingportion; and wherein the length L₂ of each of the first and secondconnecting portions is within a range of 0.04 mm to 0.5 mm.
 32. Thequartz crystal unit according to claim 26; wherein the overall length,of the quartz crystal resonator is within a range of 1.02 to 1.95 mm;wherein the first vibrational portion of each of the first and secondtuning fork arms has a first length; wherein the second vibrationalportion of each of the first and second tuning fork arms has a secondlength; wherein the second length of second vibrational portion of eachof the first and second tuning fork arms is less than the first lengthof the first vibrational portion of the corresponding one of the firstand second tuning fork arms; wherein the base portion has a first baseportion including a first length and a first width, a second baseportion including a second length less than or equal to the first lengthand a second width equal to the first width, and a third base portionconnecting the first base portion to the second base portion andincluding a third length less than the first length and a third widthless than each of the first and second widths, two cut portions beingformed between the first and second base portions, each of the first andsecond tuning fork arms being connected to the first base portion of thebase portion; wherein the first mounting arm is connected to the secondbase portion of the base portion through the first connecting portionand the second mounting arm is connected to the second base portion ofthe base portion through the second connecting portion so that thesecond base portion of the base portion and the first and secondconnecting portions have a concave shape or a U-shape with the first andsecond mounting arms; wherein each of the first and second connectingportions has a width W_(S) within a range of 0.015 mm to 0.14 mm and alength L₂ within a range of 0.04 mm to 0.5 mm; wherein the first lengthof the first base portion of the base portion is greater than the widthW_(S) of each of the first and second connecting portions; and whereinthe second width of the second vibrational portion of each of the firstand second tuning fork arms is greater than the width W_(S) of each ofthe first and second connecting portions.
 33. The quartz crystal unitaccording to claim 26; wherein the base portion has a first base portionincluding a first length, a second base portion, and a third baseportion connecting the first base portion to the second base portion,two cut portions being formed between the first and second baseportions, each of the first and second tuning fork arms being connectedto the first base portion of the base portion; wherein the firstmounting arm is connected to the second base portion of the base portionthrough the first connecting portion and the second mounting arm isconnected to the second base portion of the base portion through thesecond connecting portion so that the second base portion of the baseportion and the first and second connecting portions have a concaveshape or a U-shape with the first and second mounting arms; wherein theoverall length of the quartz crystal resonator is within a range of 1.02mm to 1.95 mm; wherein the first width of the first vibrational portionof each of the first and second tuning fork arms is defined by W and aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and die first vibrational portion of the second tuningfork arm is defined by W₄; wherein the sum of the first width W of thefirst vibrational portion of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm is defined by a width W₅=2W+W₄; whereinthe width W₅ is less than the width of the base portion; wherein thefirst mounting arm is connected to the first connecting portion andadjoins the first tuning fork arm, each of the first connecting portionand the first tuning fork arm having an outer edge; wherein a distancein the length direction of the first connecting portion measured fromthe outer edge of the first connecting portion to the outer edge of thefirst tuning fork arm is defined by L₄; wherein the distance L₄ iswithin a range of 0.012 mm 0.38 mm; wherein the first vibrationalportion of each of the first and second tuning fork arms has a firstlength and the second vibrational portion of each of the first andsecond tuning fork arms has a second length; wherein the second lengthof the second vibrational portion of each of the first and second tuningfork arms is less than the first length of the first vibrational portionof the corresponding one of the first and second tuning fork arms andgreater than each of the first length of the first base portion of thebase portion and the second width of the second vibrational portion ofthe corresponding one of the first and second tuning fork arms; andwherein the second width of the second vibrational portion of each ofthe first and second tuning fork arms is greater than or equal to thespaced-apart distance W₄ between the first vibrational portion of thefirst tuning fork arm and the first vibrational portion of the secondtuning fork arm.
 34. A quartz crystal unit comprising: a case; a lid;and a quartz crystal resonator having a base portion, and first andsecond tuning fork arms connected to the base portion, the base portionhaving a length greater than a spaced-apart distance between the firsttuning fork arm and the second tuning fork arm, each of the first andsecond tuning fork arms having a width less than 0.18 mm, and aplurality of different widths including a first width and a second widthgreater than the first width, and opposite main surfaces, at least onegroove being formed in at least one of the opposite main surfaces ofeach of the first and second tuning fork arms so that a length of the atleast one groove formed in the at least one of the opposite mainsurfaces of each of the first and second tuning fork arms is within arange of 0.45 mm to 1.25 mm; wherein the case has an open end, thequartz crystal resonator is housed in the case, and the lid is connectedto the case and closes the open end of the case.
 35. The quartz crystalunit according to claim 34; wherein each of the first and second tuningfork arms comprises a first vibrational portion having the first width,and a second vibrational portion having the second width greater thanthe first width, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface; wherein the at least onegroove comprises a groove having a width W₂ and formed in at least oneof the first and second main surfaces of the first vibrational portionof each of the first and second tuning fork arms so that a length of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms is within a range of 0.45 mm to 1.25 mm, a ratio of the widthW₂ of the groove formed in the at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms and the first width W of the first vibrationalportion of the corresponding one of the first and second tuning forkarms being defined by W₂₁=W₂/W, and so that the ratio W₂₁ is within arange of 0.35 to 0.95, the first vibrational portion of each of thefirst and second tuning fork arms having a first length and the secondvibrational portion of each of the first and second tuning fork armshaving a second length; wherein the second length of the secondvibrational portion of each of the first and second tuning fork arms isless than the first length of the first vibrational portion of thecorresponding one of the first and second tuning fork arms; wherein thecase has a mounting portion; wherein the base portion is mounted on themounting portion of the case; and wherein the lid is connected to thecase to cover an open end of the case.
 36. The quartz crystal unitaccording to claim 35; wherein the base portion has a first base portionincluding a first length, a second base portion, a third base portionconnecting the first base portion to the second base portion and two cutportions formed between the first and second base portions, the thirdbase portion having a third length, each of the first and second tuningfork arms being connected to the first base portion of the base portion;wherein the second base portion of the base portion is mounted on themounting portion of the case; wherein the second length of the secondvibrational portion of each of the first and second tuning fork arms isgreater than the first length of the first base portion of the baseportion; wherein the second width of the second vibrational portion ofeach of the first and second tuning fork arms is greater than aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and the first vibrational portion of the second tuningfork arm; wherein the first length of the first base portion is within arange of 0.1 mm to 0.23 mm and greater than the third length of thethird base portion; wherein a width of the base portion is within arange of 0.15 mm to 0.53 mm; and wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms iswithin a range of 0.12 mm to 0.26 mm and greater than a spaced-apartdistance between the second vibrational portion of the first tuning forkarm and the second vibrational portion of the second tuning fork arm.37. The quartz crystal unit according to claim 35; wherein the baseportion has a first base portion, a second base portion including asecond length, and a third base portion connecting the first baseportion to the second base portion, two cut portions being formedbetween the first and second base portions, each of the first and secondtuning fork arms being connected to the first base portion of the baseportion; wherein the second base portion of the base portion is mountedon the mounting portion of the case; and wherein the second length ofthe second vibrational portion, of each of the first and second tuningfork arms is greater than the second length of the second base portionof the base portion.
 38. The quartz crystal unit according to claim 34;wherein the case has a mounting portion; wherein the base portion ismounted on the mounting portion of the case; wherein the lid isconnected to the case to cover an open end of the case; wherein each ofthe first and second tuning fork arms comprises a first vibrationalportion having the first width, and a second vibrational portion havingthe second width greater than the first width so that the firstvibrational portion is formed between the second vibrational portion andthe base portion, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface, each of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms having a central linear portion;wherein the at least one groove comprises a groove formed in the centrallinear portion of at least one of the first and second main surfaces ofthe first vibrational portion of each of the first and second tuningfork arms so that a width of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is greater than a distance in the width direction of the groovemeasured from an outer edge of the groove to an outer edge of thecorresponding one of the first and second tuning fork arms, the firstvibrational portion of each of the first and second tuning fork armshaving a first length and the second vibrational portion of each of thefirst and second tuning fork arms having a second length; wherein thesecond length of the second vibrational portion of each of the first andsecond tuning fork arms is less than the first length of the firstvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than the second width of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms; and wherein the second width of the second vibrationalportion of each of the first and second tuning fork arms is greater thanor equal to a spaced-apart distance between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm.
 39. The quartz crystal unit according toclaim 34; wherein the base portion has a first base portion including afirst length, a second base portion including a second length and twocut portions formed between the first and second base portions, each ofthe first and second tuning fork arms being connected to the first baseportion of the base portion; wherein the case has a mounting portion;wherein the second base portion of the base portion is mounted on themounting portion of the case; wherein the lid is connected to the caseto cover an open end of the case; wherein each of the first and secondtuning fork arms comprises a first vibrational portion having the firstwidth, and a second vibrational portion having the second width greaterthan the first width so that the first vibrational portion is formedbetween the second vibrational portion and the base portion, the firstvibrational portion of each of the first and second tuning fork armshaving the opposite main surfaces including a first main surface and asecond main surface; wherein the at least one groove comprises a grooveformed in at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms so that a length of the groove formed in the at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms is within a range of 0.45 mm to1.25 mm, the first width of the first vibrational portion of each of thefirst and second tuning fork arms being greater than 0.025 mm and lessthan 0.175 mm, a first distance in the width direction of the groovemeasured from a first outer edge of the groove to a first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that a width of the groove formed in the at least one ofthe first and second main surfaces of the first vibrational portion ofeach of the first and second tuning fork arms is greater than each ofthe first distance W₁ and the second distance W₃, the first vibrationalportion of each of the first and second tuning fork arms having a firstlength and the second vibrational portion of each of the first andsecond tuning fork arms having a second length; wherein the secondlength of the second vibrational portion of each of the first and secondtuning fork arms is less than the first length of the first vibrationalportion of the corresponding one of the first and second tuning forkarms and greater than each of the first length of the first base portionof the base portion and the second length of the second base portion ofthe base portion; and wherein the second width of the second vibrationalportion of each of the first and second tuning fork arms is greater thana spaced-apart distance between the second vibrational portion of thefirst tuning fork arm and the second vibrational portion of the secondtuning fork arm.
 40. The quartz crystal unit according to claim 39;wherein the second width of the second vibrational portion of each ofthe first and second tuning fork arms is within a range of 0.12 mm to0.26 mm; wherein the first width of the first vibrational portion ofeach of the first and second tuning fork arms is defined by W and aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and the first vibrational portion of the second tuningfork arm is defined by W₄, wherein the sum of the first width W of thefirst vibrational portion of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm is defined by a width W₅=2W+W₄; andwherein the width W₅ is within a range of 0.15 mm to 0.52 mm.
 41. Thequartz crystal unit according to claim 39; wherein the spaced-apartdistance between the second vibrational portion of the first tuning forkarm and the second vibrational portion of the second tuning fork arm iswithin a range of 0.042 mm to 0.062 mm.
 42. The quartz crystal unitaccording to claim 34; wherein each of the first and second tuning forkarms comprises a first vibrational portion having the first width, and asecond vibrational portion having the second width greater than thefirst width so that the first vibrational portion is formed between thesecond vibrational portion and the base portion, the first vibrationalportion of each of the first and second tuning fork arms having theopposite main surfaces including a first main surface and a second mainsurface; wherein the at least one groove comprises a groove formed in atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms so that a widthof the groove formed in the at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms is defined by W₂ and greater than a distance inthe width direction of the groove measured from an outer edge of thegroove to an outer edge of the corresponding one of the first and secondtuning fork arms; wherein the base portion has a first base portionincluding a first length, a second base portion and a third base portionconnecting the first base portion to the second base portion, two cutportions being formed between the first and second base portions, eachof the first and second tuning fork arms being connected to the firstbase portion of the base portion; wherein the case has a mountingportion; wherein the second base portion of the base portion is mountedon the mounting portion of the case; wherein the lid is connected to thecase to cover an open end of the case; wherein the first width of thefirst vibrational portion of each of the first and second tuning forkarms is greater than 0.025 mm and less than 0.175 mm, the firstvibrational portion of each of the first and second tuning fork armshaving a first length and the second vibrational portion of each of thefirst and second tuning fork arms having a second length; and whereinthe second length of the second vibrational portion of each of the firstand second tuning fork arms is less than the first length of the firstvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than the first length of the first baseportion of the base portion.
 43. The quartz crystal unit according toclaim 42; wherein a length of the groove formed in the at least one ofthe first and second main surfaces of the first vibrational portion ofeach of the first and second tuning fork arms is within a range of 0.45mm to 1.25 mm; wherein the first width of the first vibrational portionof each of the first and second tuning fork arms is defined by W and aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and the first vibrational portion of the second tuningfork arm is defined by W₄; wherein the sum of the first width W of thefirst vibrational portion of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm is defined by a width W₅=2W+W₄; whereinthe width W₅ is within a range of 0.15 mm 0.52 mm; wherein the baseportion has the second base portion including a second length; andwherein, the second length of the second vibrational portion of each ofthe first and second tuning fork arms is greater than the second lengthof the second base portion of the base portion.
 44. The quartz crystalunit according to claim 43; wherein each of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms has a central linear portion; wherein the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is fainted in the central linear portion of at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms so that a length of the grooveformed in the central linear portion of the at least one of the firstand second main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is within a range of 0.45 mm to 1.25mm, a ratio of the width W₂ of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms and the first width W of the first vibrational portion of thecorresponding one of the first and second tuning fork arms being definedby W₂₁=W₂/W, and so that the ratio W₂₁ is within a range of 0.35 to0.95; wherein the base portion has the first base portion including afirst width, the second base portion including a second width and thethird base portion in a third width less than each of the first andsecond widths; and wherein the width W₅ is less than or equal to a widthof the base portion, the width of the base portion being within a rangeof 0.15 mm to 0.53 mm.
 45. The quartz crystal unit according to claim43; wherein each of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms hasa central linear portion; wherein the at least one groove comprises agroove formed in the central linear portion of at least one of the firstand second main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms so that a length of the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is within a range of 0.45 mm to 1.25mm; and wherein the second width of the second vibrational portion ofeach of the first and second tuning fork arms is within a range of 0.12mm to 0.26 mm and greater than a spaced-apart distance between thesecond vibrational portion of the first tuning fork arm and the secondvibrational portion of the second tuning fork arm.
 46. The quartzcrystal unit according to claim 43; wherein each of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms has a central linear portion; wherein the atleast one groove comprises a groove formed in the central linear portionof at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms sothat a width of the groove formed in the central linear portion of theat least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms isless than 0.07 mm and greater than each of a first distance in the widthdirection of the groove measured from a first outer edge of the grooveto a first outer edge of the corresponding one of the first and secondtuning fork arms and a second distance in the width direction of thegroove measured from a second outer edge opposite the first outer edgeof the groove to a second outer edge opposite the first outer edge ofthe corresponding one of the first and second tuning fork arms; andwherein the second width of the second vibrational portion of each ofthe first and second tuning fork arms is greater than the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning forkarm.
 47. The quartz crystal unit according to claim 42; wherein thesecond width of the second vibrational portion of each of the first andsecond tuning fork arms is less than the second length of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than a spaced-apart distance between thesecond vibrational portion of the first tuning fork arm and the secondvibrational portion of the second tuning fork arm; wherein the baseportion has the second base portion including a second length; andwherein the second length of the second vibrational portion of each ofthe first and second tuning fork arms is greater than the second lengthof the second base portion of the base portion.
 48. The quartz crystalunit according to claim 47; wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms iswithin a range of 0.12 mm to 0.26 mm; wherein the spaced-apart distancebetween the second vibrational portion of the first tuning fork arm andthe second vibrational portion of the second tuning fork arm is within arange of 0.042 mm to 0.062 mm; and wherein the second width of thesecond vibrational portion of each of the first and second tuning forkarms is greater than a spaced-apart distance between the firstvibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm.
 49. The quartzcrystal unit according to claim 47; wherein the base portion has thefirst base portion including a first width, the second base portionincluding a second width and the third base portion including a thirdwidth less than each of the first and second widths; wherein the firstlength of the first base portion of the base portion is within a rangeof 0.1 mm to 0.23 mm; wherein a width of the base portion is within arange of 0.15 mm to 0.53 mm; and wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms isgreater than a spaced-apart distance between the first vibrationalportion of the first to fork arm and the first vibrational portion ofthe second tuning fork arm.
 50. The quartz crystal unit according toclaim 34; wherein each of the first and second tuning fink armscomprises a first vibrational portion having the first width, and asecond vibrational portion having the second width greater than thefirst width so that the first vibrational portion is formed between thesecond vibrational portion and the base portion, the first vibrationalportion of each of the first and second tuning fork arms having theopposite main surfaces including a first main surface and a second mainsurface, each of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork armshaving a central linear portion; wherein the at least one groovecomprises a groove formed in the central linear portion of at least oneof the first and second main surfaces of the first vibrational portionof each of the first and second tuning fork arms so that a length ofgroove formed in the central linear portion of the at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms is within a range of 0.45 mm to1.25 mm, a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that a width of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is greater than each of the first distance W₁ and the seconddistance W₃ and the first distance W₁ is greater than or equal to thesecond distance W₃; wherein the base portion has a first base portion, asecond base portion and two cut portions formed between the first andsecond base portions, each of the first and second tuning fork armsbeing connected to the first base portion of the base portion; whereinthe case has a mounting portion; wherein the second base portion of thebase portion is mounted on the mounting portion of the case; wherein thelid is connected to the case to cover an open end of the case, thesecond vibrational portion of each of the first and second tuning forkarms having a second length; and wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms isless than the second length of the second vibrational portion of thecorresponding one of the first and second tuning fork arms and greaterthan a spaced-apart distance between the second vibrational portion ofthe first tuning fork arm and the second vibrational portion of thesecond tuning fork arm.
 51. The quartz crystal unit according to claim50; wherein the first base portion of the base portion has a firstlength and a first width; wherein the base portion has a third baseportion connecting the first base portion to the second base portion andincluding a third width less than the first width of the first baseportion; wherein the second length of the second vibrational portion ofeach of the first and second tuning fork arms is greater than the firstlength of the first base portion of the base portion; and wherein thesecond width of the second vibrational portion of each of the first andsecond tuning fork arms is greater than a spaced-apart distance betweenthe first vibrational portion of the first tuning form arm and the firstvibrational portion of the second tuning fork arm.
 52. The quartzcrystal unit according to claim 50; wherein the second base portion ofthe base portion has a second length, the first vibrational portion ofeach of the first and second tuning fork arms having a first length; andwherein the second length of the second vibrational portion of each ofthe first and second tuning fork arms is less than the first length ofthe first vibrational portion of the corresponding one of the first andsecond tuning fork arms and greater than the second length of the secondbase portion of the base portion.
 53. The quartz crystal unit accordingto claim 34; wherein each of the first and second tuning fork armscomprises a first vibrational portion having the first width, and asecond vibrational portion having the second width greater than thefirst width, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface; wherein the at least onegroove comprises only one groove formed in at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms so that a width W₂ of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is greater than each of a first distance W₁ in the width directionof the groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms and a second distance W₃ in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms, the firstwidth of the first vibrational portion of each of the first and secondtuning fork arms being defined by W and a spaced-apart distance betweenthe first vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm being defined by W₄;wherein the sum of the first width W of the first vibrational portion ofeach of the first and second tuning fork arms and the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning fork armis defined by a width W₅=2W+W₄; wherein the width W₅ is within a rangeof 0.15 mm to 0.52 mm, the second vibrational portion of each of thefirst and second tuning fork arms having a second length; and whereinthe second width of the second vibrational portion of each of the firstand second tuning fork arms is less than the second length of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than the spaced-apart distance W₄ betweenthe first vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm.
 54. The quartzcrystal unit according to claim 53; wherein the base portion has a firstbase portion including a first length and a first width, a second baseportion including a second length and a second width, and a third baseportion connecting the first base portion to the second base portion,the third base portion having a third width less than each of the firstand second widths, each of the first and second tuning fork arms beingconnected to the first base portion of the base portion; wherein thecase has a mounting portion; wherein the second base portion of the baseportion is mounted on the mounting portion of the case; wherein the lidis connected to the case to cover an open end of the case, the firstvibrational portion of each of the first and second tuning fork armshaving a first length; wherein the second length of the secondvibrational portion of each of the first and second tuning fork arms isless than the first length of the first vibrational portion of thecorresponding one of the first and second tuning fork arms and greaterthan each of the first length of the first base portion of the baseportion and the second length of the second base portion of the baseportion; wherein the first distance W₁ is greater than or equal to thesecond distance W₃; wherein the first length of the first base portionof the base portion is within a range of 0.1 mm to 0.23 mm; and whereinthe second width of the second vibrational portion of each of the firstand second tuning fork arms is within a range of 0.12 mm to 0.26 mm andgreater than a spaced-apart distance between the second vibrationalportion of the first tuning fork arm and the second vibrational portionof the second tuning fork arm.
 55. The quartz crystal unit according toclaim 34; wherein each of the first and second tuning fork armscomprises a first vibrational portion having the first width W, and asecond vibrational portion having the second width We greater than thefirst width W, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface, each of the first andsecond main surface of the first vibrational portion of each of thefirst and second tuning fork arms having a central linear portion;wherein the at least one groove comprises a groove having a width W₂ andformed in the central linear portion of at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms so that a length of the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is within a range of 0.45 mm to 1.25mm, a first distance in the width direction of the groove measured froma first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, a ratio of the width W₂ of the groove formed in the centrallinear portion of the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms and the first width W of the first vibrational portion of thecorresponding one of the first and second tuning fork arms being definedby W₂₁=W₂/W, and so that the width W₂ of the groove thrilled in thecentral linear portion of the at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms is greater than each of the first distance W₁and the second distance W₃ and the ratio W₂₁ is within a range of 0.35to 0.95, a spaced-apart distance between the first vibrational portionof the first tuning fork arm and the first vibrational portion of thesecond tuning fork arm being defined by W₄; wherein the sum of the firstwidth W of the first vibrational portion of each of the first and secondtuning fork arms and the spacer apart distance W₄ between the firstvibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by a widthW₅=2W+W₄; wherein the width W₅ is within a range of 0.15 mm to 0.52 mm,the first vibrational portion of each of the first and second tuningfork arms having a first length and the second vibrational portion ofeach of the first and second tuning fork arms having a second length;wherein the second length of the second vibrational portion of each ofthe first and second tuning fork arms is less than the first length ofthe first vibrational portion of the corresponding one of the first andsecond tuning fork arms; wherein the second width We of the secondvibrational portion of each of the first and second tuning fork arms isless than the second length of the second vibrational portion of thecorresponding one of the first and second tuning fork arms and greaterthan the spaced-apart distance W₄ between the first vibrational portionof the first tuning fork arm and the first vibrational portion of thesecond tuning fork arm; wherein the case has a mounting portion; whereinthe base portion is mounted on the mounting portion of the case; andwherein the lid is connected to the case to cover an open end of thecase.
 56. The quartz crystal unit according to claim 55; wherein thebase portion has a first base portion including a first length, a secondbase portion including a second length and a third base portionconnecting the first base portion to the second base portion, two cutportions being formed between the first and second base portions, eachof the first and second tuning fork arms being connected to the firstbase portion of the base portion; wherein the second base portion of thebase portion is mounted on the mounting portion of the case; wherein thesecond width We of the second vibrational portion of each of the firstand second tuning fork arms is greater than a spaced-apart distancebetween the second vibrational portion of the first tuning fork arm andthe second vibrational portion of the second tuning fork arm; andwherein the second length of the second vibrational portion of each ofthe first and second tuning fork arms is greater than each of the firstlength of the first base portion and the second length of the secondbase portion.
 57. The quartz crystal unit according to claim 56; whereinthe first length of the first base portion of the base portion is withina range of 0.1 mm to 0.23 mm; and wherein the second width We of thesecond vibrational portion of each of the first and second tuning forkarms is within a range of 0.12 mm to 0.26 mm.
 58. The quartz crystalunit according, to claim 56; wherein the width W₂ of the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is less than 0.07 mm.
 59. The quartzcrystal unit according to claim 56; wherein the spaced-apart distancebetween the second vibrational portion of the first tuning fork arm andthe second vibrational portion of the second tuning fork arm is within arange of 0.042 mm to 0.062 mm.
 60. The quartz crystal unit according toclaim 34; wherein the base portion has a first base portion, a secondbase portion and a third base portion connecting the first base portionto the second base portion, two cut portions being formed between thefirst and second base portions, each of the first and second baseportions of the base portion having a length, each of the first andsecond tuning fork arms being connected to the first base portion of thebase portion; wherein the case has a mounting portion; wherein thesecond base portion of the base portion is mounted on the mountingportion of the case; wherein the lid is connected to the case to coveran open end of the case; wherein each of the first and second tuningfork arms comprises a first vibrational portion having the first width,and a second vibrational portion having the second width greater thanthe first width, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces includingfirst and second main surfaces each having a central linear portion;wherein the at least one groove comprises a groove formed in the centrallinear portion of at least one of the first and second main surfaces ofthe first vibrational portion of each of the first and second tuningfork arms so that a width W₂ of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is greater than each of a first distance W₁ in the width directionof the groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms and a second distance W₃ in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms, the secondvibrational portion of each of the first and second tuning fork armshaving a second length; wherein the second length of the secondvibrational portion of each of the first and second tuning fork arms isgreater than the length of each of the first and second base portions ofthe base portion, and wherein the first distance W₁ is greater than orequal to the second distance W₃.
 61. The quartz crystal unit accordingto claim 60; wherein the first width of the first vibrational portion ofeach of the first and second tuning fork arms is defined by W and aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and the first vibrational portion of the second tuningfork arm is defined by W₄; wherein the sum of the first width W of thefirst vibrational portion of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm is defined by a width W₅=2W+W₄; whereinthe width W₅ is within a range of 0.15 mm to 0.52 mm; wherein the firstvibrational portion of each of the first and second tuning fork arms hasa first length; wherein the first length of the first vibrationalportion of each of the first and second tuning fork arms is greater thanthe second length of the second vibrational portion of the correspondingone of the first and second tuning fork arms; and wherein the secondwidth of the second vibrational portion of each of the first and secondtuning fork arms is less than the second length of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than the spaced-apart distance W₄ betweenthe first vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm.
 62. The quartzcrystal unit according to claim 60; wherein the first vibrationalportion of each of the first and second tuning fork arms has a firstlength; wherein the first length of the first vibrational portion ofeach of the first and second tuning fork arms is greater than the secondlength of the second vibrational portion of the corresponding one of thefirst and second tuning fork arms; and wherein the second width of thesecond vibrational portion of each of the first and second tuning forkarms is less than the second length of the second vibrational portion ofthe corresponding one of the first and second tuning fork arms andgreater than a spaced-apart distance between the second vibrationalportion of the first tuning fork arm and the second vibrational portionof the second tuning fork arm.
 63. The quartz crystal unit according toclaim 62; wherein the second width of the second vibrational portion ofeach of the first and second tuning fork arms is within a range of 0.12mm to 0.26 mm; and wherein the spaced-apart distance between the secondvibrational portion of the first tuning fork arm and the secondvibrational portion of the second tuning fork arm is within a range of0.042 mm to 0.062 mm; wherein a width of the base portion is within arange of 0.15 mm to 0.53 mm; wherein the first length of the first baseportion of the base portion is within a range of 0.1 mm to 0.23 mm;wherein the first width of the first vibrational portion of each of thefirst and second tuning fork arms is greater than 0.025 mm and less than0.175 mm; and wherein a length of the groove formed in the centrallinear portion of the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms is within a range of 0.45 mm to 1.25 mm.
 64. The quartzcrystal unit according to claim 34; wherein each of the first and secondtuning fork arms comprises a first vibrational portion having the firstwidth, and a second vibrational portion having the second width greaterthan the first width, the first vibrational portion of each of the firstand second tuning fork arms having the opposite main surfaces includinga first main surface and a second main surface, each of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms having a central linear portion;wherein the at least one groove comprises a groove formed in the centrallinear portion of at least one of the first and second main surfaces ofthe first vibrational portion of each of the first and second tuningfork arms so that a width of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is less than 0.07 mm, a first distance in the width direction ofthe groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms being defined by W₁ and a second distance in the width direction ofthe groove measured from a second outer edge opposite the first outeredge of the groove to a second outer edge opposite the first outer edgeof the corresponding one of the first and second tuning fork arms beingdefined by W₃, and so that each of the first distance W₁ and the seconddistance W₃ is less than 0.015 mm; wherein an overall length of thequartz crystal resonator is less than 2.1 mm; wherein the length of thebase portion is less than 0.5 mm; wherein at least one mounting arm isconnected to the base portion through a connecting portion and has alength within a range of 0.3 mm to 1.85 mm so that the at least onemounting arm extends in a common direction with the first and secondtuning fork arms outside the first and second tuning fork arms; whereinthe case has a mounting portion; wherein the at least one mounting armis mounted on the mounting portion of the case; wherein an electrode isdisposed on each of a surface of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms, a surface of the connecting portion, a surface of the at least onemounting arm and a surface of the mounting portion of the case; whereinthe electrode disposed on the surface of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of the first vibrational portion of one of the first and secondtuning fork arms is connected to the electrode disposed on the surfaceof the at least one mounting arm through the electrode disposed on thesurface of the connecting portion, and the electrode disposed on thesurface of the at least one mounting arm is connected to the electrodedisposed on the surface of the mounting portion of the case; and whereinthe lid is connected to the case to cover an open end of the case. 65.The quartz crystal unit according to claim 64; wherein the length of thebase portion is defined by L₁ and a length of the connecting portion isdefined by L₂; wherein the length L₁ of the base portion is greater thanthe length L₂ of the connecting portion; wherein the first vibrational,portion of each of the first and second tuning fork arms has the firstwidth defined by W and a spaced-apart distance between the firstvibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by W₄;wherein the sum of the first width W of the first vibrational portion ofeach of the first and second tuning fork arms and the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning fork armis defined by a width W₅=2W+W₄; wherein the width W₅ is less than awidth of the base portion; wherein the connecting portion is connectedto the at least one mounting arm and has an outer edge, and one of thefirst and second tuning fork arms adjoins the at least one mounting armand has an outer edge; wherein a distance in the length direction of theconnecting portion measured from the outer edge of the connectingportion to the outer edge of one of the first and second tuning forkarms is defined by L₄; and wherein the distance L₄ is within a range of0.012 mm to 0.38 mm.
 66. The quartz crystal unit according to claim 64;wherein the at least one mounting arm has first and second mounting armseach including a length within a range of 0.3 mm to 1.85 mm, wherein thefirst mounting arm is connected to the base portion through a firstconnecting portion and the second mounting arm is connected to the baseportion through a second connecting portion, each of the first andsecond mounting arms extending in a common direction with the first andsecond tuning fork arms outside the first and second tuning fork arms;wherein the case has first and second mounting portions; wherein thefirst mounting arm is mounted on the first mounting portion of the caseand the second mounting arm is mounted on the second mounting portion ofthe case, each of the first and second connecting portions having awidth W_(S) and a length L₂; and wherein the length L₂ of each of thefirst and second connecting portions is within a range of 0.04 mm to 0.5mm and greater than the width W_(S) of each of the first and secondconnecting portions.
 67. The quartz crystal unit according to claim 64;wherein the length of the base portion is within a range of 0.015 mm to0.49 mm; herein the width of the groove formed in the central linearportion of the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms is within a range of 0.038 mm to 0.05 mm; wherein each of the firstdistance W₁ and the second distance W₃ is within a range of 0.001 mm to0.012 mm; wherein the at least one mounting arm has first and secondmounting arms each including a length within a range of 0.3 mm to 1.85mm; wherein the first mounting arm is connected to the base portionthrough a first connecting portion and the second mounting arm isconnected to the base portion through a second connecting portion, eachof the first and second mounting arms extending in a common directionwith the first and second tuning fork arms outside the first and secondtuning fork arms; wherein the case has first and second mountingportions; wherein the first mounting arm is mounted on the firstmounting portion of the case and the second mounting arm is mounted onthe second mounting portion of the case; wherein the groove formed inthe central linear portion of the at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms is formed in the central linear portion of eachof the first and second main surfaces of each of the first and secondtuning fork arms; wherein a first electrode is disposed on a surface ofthe groove formed in the central linear portion of each of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms; wherein a second electrode isdisposed on a surface of each of the first and second connectingportions; wherein a third electrode is disposed on a surface of each ofthe first and second mounting arms; wherein a fourth electrode isdisposed on a surface of each of the first and second mounting portionsof the case; wherein the first electrode disposed cm the surface of thegroove formed in the central linear portion of each, of the first andsecond main surfaces of the first vibrational portion of one of thefirst and second tuning fork arms is connected to the third electrodedisposed on the surface of the first mounting arm through the secondelectrode disposed on the surface of the first connecting portion, andthe third electrode disposed on the surface of the first mounting arm isconnected to the fourth electrode disposed on the surface of the firstmounting portion of the case; and wherein the first electrode disposedon the surface of the groove formed in the central linear portion ofeach of the first and second main surfaces of the first vibrationalportion of the other of the first and second tuning fork arms isconnected to the third electrode disposed on the surface of the secondmounting arm through the second electrode disposed on the surface of thesecond connecting portion, and the third electrode disposed on thesurface of the second mounting arm is connected to the fourth electrodedisposed on the surface of the second mounting portion of the case. 68.The quartz crystal unit according to claim 64; wherein the overalllength of the quartz crystal resonator is within a range of 1.02 mm to1.95 mm; wherein the length of the base portion is within a range of0.015 mm to 0.49 mm; wherein the at least one mounting arm has a widthW₆ and adjoins one of the first and second tuning fork arms; wherein thefirst width of the first vibrational portion of one of the first andsecond tuning fork arms is defined by W; and wherein the width W₆ of theat least one mounting arm is less than or equal to the first width W ofthe first vibrational portion of one of the first and second tuning forkarms.
 69. The quartz crystal unit according to claim 64; wherein thebase portion has a first base portion including a first length, a secondbase portion and a third base portion connecting the first base portionto the second base portion, two cut portions being formed between thefirst and second base portions, each of the first and second tuning forkarms being connected to the first base portion of the base portion;wherein the at least one mounting arm has first and second mountingarms; wherein the first mounting arm is connected to the second baseportion of the base portion through a first connecting, portion and thesecond mounting arm is connected to the second base portion of the baseportion through a second connecting portion so that the second baseportion of the base portion and the first and second connecting portionshave a concave shape or a U-shape with the first and second mountingarms, each of the first and second mounting arms having a length L₃;wherein the length L₃ of each of the first and second mounting arms iswithin a range of 0.3 mm to 1.85 mm; wherein the case has first andsecond mounting portions; wherein the first mounting arm is mounted onthe first mounting portion of the case and the second mounting arm ismounted on the second mounting portion of the case, the firstvibrational portion of each of the first and second tuning fork armshaving a first length and the second vibrational portion of each of thefirst and second tuning fork arms having a second length; wherein thesecond length of second vibrational portion of each of the first andsecond tuning fork arms is less than the first length of the firstvibrational portion of the corresponding one of the first and secondtuning fork arms and greater than each of the second width of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms and the first length of the first base portion of thebase portion; wherein the second width of the second vibrational portionof each of the first and second tuning fork arms is greater than aspaced-apart distance between the second vibrational portion of thefirst tuning fork arm and the second vibrational portion of the secondtuning fork arm; wherein the overall length of the quartz crystalresonator is within a range of 1.02 mm to 1.95 mm; and wherein thelength of the base portion is within a range of 0.015 mm to 0.49 mm. 70.The quartz crystal unit according to claim 64; wherein the first widthof the first vibrational portion of each of the first and second tuningfork arms is defined by W and a spaced-apart distance between the firstvibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by W₄;wherein the sum of the first width W of the first vibrational portion ofeach of the first and second tuning fork arms and the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning fork armis defined by a width W₅=2W+W₄; wherein the width W₅ is within a rangeof 0.15 mm to 0.52 mm; wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms isgreater than the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm; wherein the base portion has a first baseportion, a second base portion including a second length and two cutportions formed between the first and second base portions, each of thefirst and second tuning fork arms being connected to the first baseportion of the base portion; wherein the at least one mounting arm hasfirst and second mounting arms; wherein the first mounting arm isconnected to the second base portion of the base portion through a firstconnecting portion and the second mounting arm is connected to thesecond base portion of the base portion through a second connectingportion so that the second base portion of the base portion and thefirst and second connecting portions have a concave shape or a U-shapewith the first and second mounting arms, each of the first and secondmounting arms having a length L₃; wherein the length L₃ of each of thefirst and second mounting arms is within a range of 0.3 mm to 1.85 mm;wherein the case has first and second mounting portions; wherein thefirst mounting arm is mounted on the first mounting portion of the caseand the second mounting arm is mounted on the second mounting portion ofthe case, the second vibrational portion of each of the first and secondtuning fork arms having a second length; and wherein the second lengthof second vibrational portion of each of the first and second tuningfork arms is greater than the second length of the second base portionof the base portion.
 71. The quartz crystal unit according to claim 64;wherein the length of the base portion is defined by L₁; wherein thelength of the connecting portion is defined by L₂; wherein the length L₁of the base portion is greater than the length L₂ of the connectingportion; wherein a difference of the length L₁ of the base portion andthe length L₂ of the connecting portion is defined by L₁₂=L₁−L₂; whereinthe difference L₁₂ is within a range of 0.1 mm to 0.32 mm; wherein thefirst vibrational portion of each of the first and second tuning forkarms has a first length within a range of 0.32 mm to 0.72 mm; whereinthe second vibrational portion of each of the first and second tuningfork arms has a second length within a range of 0.17 mm to 0.38 mm;wherein the at least one mounting arm has first and second mountingarms; wherein the first mounting arm is connected to the base portionthrough a first connecting portion and second mounting arm is connectedto the base portion through a second connecting portion, each of thefirst and second mounting arms extending in a common direction with thefirst and second tuning fork arms outside the first and second tuningfork arms; wherein the case has first and second mounting portions;wherein the first mounting arm is mounted on the first mounting portionof the case and the second mounting arm is mounted on the secondmounting portion of the case, the first mounting arm adjoining one ofthe first and second tuning fork arms, the second mounting arm adjoiningthe other of the first and second tuning fork arms, a spaced-apartdistance between the first vibrational portion of one of the first andsecond tuning fork arms and the first mounting arm being defined by afirst distance, a spaced-apart distance between the first vibrationalportion of the other of the first and second tuning fork arms and thesecond mounting arm being defined by a second distance and wherein thesecond width of the second vibrational portion of each of the first andsecond tuning fork arms is within a range of 0.12 mm to 0.26 mm andgreater than each of the first distance and the second distance.
 72. Thequartz crystal unit according, to claim 34; wherein each of the firstand second tuning fork arms each having the first width and the secondwidth greater than the first width has the opposite main surfacesincluding a first main surface and a second main surface, each of thefirst and second main surfaces of each of the first and second tuningfork arms having a central linear portion; wherein the at least onegroove comprises a groove formed in the central linear portion of atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms so that a width of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms is less than0.07 mm, a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that each of the first distance and the second distance W₃is less than 0.015 mm and the width of the groove formed in the centrallinear portion of the at least one of the first and second main surfacesof each of the first and second tuning fork arms is greater than orequal to each of the first distance W₁ and the second distance W₃;wherein an overall length of the quartz crystal resonator is within arange of 1.02 mm to 1.95 mm; wherein the length of the base portion isdefined by L₁ and less than 0.5 mm and a width of the base portion isless than 0.55 mm; wherein at least one mounting arm is connected to thebase portion through a connecting portion so that the at least onemounting arm extends in a common direction with the first and secondtuning fork arms outside the first and second tuning fork arms, the atleast one mounting arm having a first arm portion including a firstwidth and a second arm portion including a second width greater than thefirst width; wherein the case has a mounting portion; wherein the secondarm portion of the at least one mounting arm is mounted on the mountingportion of the case, the at least one mounting DIM having a length;wherein the length of the at least one mounting arm is within a range of0.3 mm 1.85 mm; and wherein the lid is connected to the case to cover anopen end of the case.
 73. The quartz crystal unit according to claim 72;wherein the length L₁ of the base portion is within a range of 0.015 mmto 0.49 mm and the width of the base portion is within a range of 0.15mm to 0.53 mm; wherein the width of the at least one mounting arm iswithin a range of 0.08 mm to 0.4 mm and greater than the width of eachof the first and second tuning fork arms; wherein a length of theconnecting portion is defined by L₂; and wherein the length L₁ of thebase portion is greater than the length L₂ of the connecting portion.74. The quartz crystal unit according to claim 72; wherein the width ofthe at least one mounting arm is defined by W₆; wherein the first widthof each of the first and second tuning fork arms is defined by W;wherein a ratio of the width W₆ of the at least one mounting arm and thefirst width W of each of the first and second tuning fork arms isdefined by W₆₁=W₆/W; wherein the ratio W₆₁ is within a range of 1.1 to3.8; and wherein the length of the base portion is within a range of0.015 mm to 0.49 mm and the width of the base portion is within a rangeof 0.15 mm to 0.53 mm.
 75. The quartz crystal unit according to claim72; wherein the width of the base portion is within a range of 0.15 mmto 0.53 mm; wherein a length of the connecting portion is defined by L₂;wherein the length L₂ of the connecting portion is within a range of0.04 mm to 0.5 mm; wherein a difference of the length L₁ of the baseportion and the length L₂ of the connecting portion is defined byL₁₂=L₁−L₂; and wherein the difference L₁₂ is within a range of −0.1 mmto 0.32 mm.
 76. The quartz crystal unit according to claim 72; whereinthe at least one mounting arm has first and second mounting arms;wherein the first mounting arm is connected to the base portion througha first connecting portion and the second mounting arm is connected tothe base portion through a second connecting portion so that each of thefirst and second mounting arms extends in a common direction with thefirst and second tuning fork arms outside the first and second tuningfork arms; wherein the case has first and second mounting portions;wherein the first mounting arm is mounted on the first mounting portionof the case and the second mounting arm is mounted on the secondmounting portion of the case; wherein each of the first and secondtuning fork arms has a first side surface and a second side surfaceopposite the first side surface; wherein a first electrode is disposedon a surface of the groove formed in the central linear portion of atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms; wherein a second electrode is disposed on eachof the first and second side surfaces of each of the first and secondtuning fork arms; wherein a third electrode is disposed on a surface ofeach of the first and second connecting portions; wherein a fourthelectrode is disposed on a surface of each of the first and secondmounting arms; wherein a fifth electrode is disposed on a surface ofeach of the first and second mounting portions of the case; wherein eachof sixth and seventh electrodes is disposed on a surface of the baseportion; wherein the first electrode disposed on the surface of thegroove formed in the central linear portion of the at least one of thefirst and second main surfaces of the first tuning fork arm is connectedto the second electrode disposed on each of the first and second sidesurfaces of the second tuning fork arm so that the first electrode ofthe first tuning fork arm is connected to the fourth electrode disposedon the surface of one of the first and second mounting arms through thesixth electrode disposed on the surface of the base portion and thethird electrode disposed on the surface of one of the first and secondconnecting portions, and the fourth electrode disposed on the surface ofone of the first and second mounting arms is connected to the fifthelectrode disposed on the surface of one of the first and secondmounting portions of the case; and wherein the first electrode disposedon the surface of the groove formed in the central linear portion of theat least one of the first and second main surfaces of the second tuningfork arm is connected to the second electrode disposed on each of thefirst and second side surfaces of the first tuning fork arm so that thefirst electrode of the second tuning fork arm is connected to the fourthelectrode disposed on the surface of the other of the first and secondmounting arms through the seventh electrode disposed on the surface ofthe base portion and the third electrode disposed on the surface of theother of the first and second connecting portions, and the fourthelectrode disposed on the surface of the other of the first and secondmounting arms is connected to the fifth electrode disposed on thesurface of the other of the first and second mounting portions of thecase.
 77. A quartz crystal oscillator comprising: a quartz crystaloscillating circuit having a plurality of capacitors, a resistor, and aquartz crystal unit comprised of a case, a lid and a quartz crystalresonator having a base portion, and first and second tuning fork armsconnected to the base portion, the base portion having a length greaterthan a spaced-apart distance between the first tuning fork arm and thesecond tuning fork arm, each of the first and second tuning fork armshaving a width less than 0.18 mm, and a plurality of different widthsincluding a first width and a second width greater than the first width,and opposite main surfaces, at least one groove being formed in at leastone of the opposite main surfaces of each of the first and second tuningfork arms so that a length of the at least one groove formed in the atleast one of the opposite main surfaces of each of the first and secondtuning fork arms is within a range of 0.45 mm to 1.25 mm, wherein thecase has an open end, the quartz crystal resonator is housed in thecase, and the lid is connected to the case and closes the open end ofthe case.
 78. The quartz crystal oscillator according to claim 77;wherein each of the first and second tuning fork arms comprises a firstvibrational portion having the first width, and a second vibrationalportion having the second width greater than the first width, the first,vibrational portion of each of the first and second tuning fork armshaving the opposite main surfaces including a first main surface and asecond main surface; wherein the at least one groove comprises a groovehaving a width W₂ and formed in at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms so that a length of the groove formed in the atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms is within arange of 0.45 mm to 1.25 mm, a ratio of the width W₂ of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms and the first width W of the first vibrational portion of thecorresponding one of the first and second tuning fork arms being definedby W₂₁=W₂/W, a first distance in the width direction of the groovemeasured from a first outer edge of the groove to a first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃, and so that the ratio W₂₁ is within a range of 0.35 to 0.95, andthe width W₂ of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is greater than each of the firstdistance W₁ and the second distance W₃, the first vibrational portion ofeach of the first and second tuning fork arms having a first length andthe second vibrational portion of each of the first and second tuningfork arms having a second length; wherein the second length of thesecond vibrational portion of each of the first and second tuning forkarms is less than the first length of the first vibrational portion ofthe corresponding one of the first and second tuning fork arms; whereinthe base portion has a first base portion including a first length, asecond base portion, and a third base portion connecting the first baseportion to the second base portion, two cut portions being formedbetween the first and second base portions, each of the first and secondtuning fork arms being connected to the first base portion of the baseportion; wherein the case has a mounting portion; wherein the secondbase portion of the base portion is mounted on the mounting portion ofthe case; wherein the lid is connected to the case to cover an open endof the case; wherein the second length of the second vibrational portionof each of the first and second tuning fork arms is greater than thefirst length of the first base portion of the base portion; and whereinthe second width of the second vibrational portion of each of the firstand second tuning fork arms is greater than a spaced-apart distancebetween the first vibrational portion of the first tuning fork arm andthe first vibrational portion of the second tuning fork arm.
 79. Thequartz crystal oscillator according to claim 78; wherein the firstlength of the first base portion of the base portion is within a rangeof 0.1 mm to 0.23 mm; wherein a width of the base portion is within arange of 0.15 mm to 0.53 mm; and wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms iswithin a range of 0.12 mm to 0.26 mm and greater than a spaced-apartdistance between the second vibrational portion of the first tuning forkarm and the second vibrational portion of the second tuning fork arm.80. The quartz crystal oscillator according to claim 77; wherein each ofthe first and second tuning fork arms comprises a first vibrationalportion having the first width, and a second vibrational portion havingthe second width greater than the first width so that the firstvibrational portion is formed between the second vibrational portion andthe base portion, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface; wherein the at least onegroove comprises a groove formed in at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms so that a width of the groove formed in the atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms is greater thaneach of a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms and a seconddistance in the width direction of the groove measured from a secondouter edge opposite the first outer edge of the groove to a second outeredge opposite the first outer edge of the corresponding one of the firstand second tuning fork arms; wherein the base portion has a first baseportion including a first length, a second base portion and a third baseportion connecting the first base portion to the second base portion,two cut portions being formed between the first and second baseportions, each of the first and second tuning fork arms being connectedto the first base portion of the base portion; wherein the case has amounting portion; wherein the second base portion of the base portion ismounted on the mounting portion of the case; wherein the lid isconnected to the case to cover an open end of the case; wherein thefirst width of the first vibrational portion of each of the first andsecond tuning fork arms is greater than 0.025 mm and less than 0.175 mm,the first vibrational portion of each of the first and second tuningfork arms having a first length and the second vibrational portion ofeach of the first and second tuning fork arms having a second length;and wherein the second length of the second vibrational portion of eachof the first and second tuning fork arms is less than the first lengthof the first vibrational portion of the corresponding one of the firstand second tuning fork arms and greater than the first length of thefirst base portion of die base portion.
 81. The quartz crystaloscillator according to claim 80; wherein the base portion has thesecond base portion including a second length; wherein the second lengthof the second vibrational portion of each of the first and second tuningfork arms is greater than the second length of the second base portionof the base portion; wherein a length of the groove formed in the atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms is within arange of 0.45 mm to 1.25 mm; wherein the first width of the firstvibrational portion of each of the first and second tuning fork arms isdefined by W and a spaced-apart distance between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm defined by W₄; wherein the sum of thefirst width W of the first vibrational portion of each of the first andsecond tuning fork arms and the spaced-apart distance W₄ between thefirst vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by a widthW₅=2W+W₄; and wherein the width W₅ is within a range of 0.15 mm to 0.52mm.
 82. The quartz crystal oscillator according to claim 81; whereineach of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms has a centrallinear portion; wherein the groove formed in the at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms is formed in the central linearportion of at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms so that a length of the groove formed in the central linear portionof the at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms iswithin a range of 0.45 mm to 1.25 mm, a ratio of the width W₂ of thegroove formed in the central linear portion of the at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms and the first width W of thefirst vibrational portion of the corresponding one of the first andsecond tuning fork arms being defined by W₂₁+W₂/W, and so that the ratioW₂₁ is within a range of 0.35 to 0.95; wherein the base portion has awidth within a range of 0.15 mm to 0.53 mm; and wherein the width W₅ isless than the width of the base portion.
 83. The quartz crystaloscillator according to claim 81; wherein each of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms has a central linear portion; wherein the atleast one groove comprises a groove formed in the central linear portionof at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms sothat a length of the groove formed in the central linear portion of theat least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms iswithin a range of 0.45 mm to 1.25 mm; and wherein the second width ofthe second vibrational portion of each of the first and second tuningfork arms is within a range of 0.12 mm to 0.26 mm and greater than aspaced-apart distance between the second vibrational portion of thefirst tuning fork arm and the second vibrational portion of the secondtuning fork arm.
 84. The quartz crystal oscillator according to claim81; wherein each of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms hasa central linear portion; wherein the at least one groove comprises agroove formed in the central linear portion of at least one of the firstand second main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms so that a width of the groove formedin the central linear portion of the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is less than 0.07 mm and greater thaneach of a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms and a seconddistance in the width direction of the groove measured from a secondouter edge opposite the first outer edge of the groove to a second outeredge opposite the first outer edge of the corresponding one of the firstand second tuning fork arms; and wherein the second width of the secondvibrational portion of each of the first and second tuning fork arms isgreater than the spaced-apart distance W₄ between the first vibrationalportion of the first tuning fork arm and the first vibrational portionof the second tuning fork arm.
 85. The quartz crystal oscillatoraccording to claim 81; wherein the base portion has the first baseportion including a first width, the second base portion including asecond width and the third base portion including a third width lessthan each of the first and second widths; wherein the first length ofthe first base portion of the base portion is within a range of 0.1 mmto 0.23 mm; wherein a width of the base portion is within a range of0.15 mm to 0.53 mm; wherein the second width of the second vibrationalportion of each of the first and second tuning fork arms is within arange of 0.12 mm to 0.26 mm and less than the second length of thesecond vibrational portion of the corresponding one of the first andsecond tuning fork arms and greater than the spaced-apart distance W₄between the first vibrational portion of the first tuning fork arm andthe first vibrational portion of the second tuning fork arm; and whereina spaced-apart distance between the second vibrational portion of thefirst tuning fork arm and the second vibrational portion of the secondtuning fork arm is within a range of 0.042 mm to 0.062 mm.
 86. Thequartz crystal oscillator according to claim 77; wherein each of thefirst and second tuning fork arms comprises a first vibrational portionhaving the first width, and a second vibrational portion having thesecond width greater than the first width, the first vibrational portionof each of the first and second tuning fork arms having the oppositemain surfaces including a first main surface and a second main surface;wherein the at least one groove comprises groove formed in at least oneof the first and second main surfaces of the first vibrational portionof each of the first and second tuning fork arms so that a width of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms is less than 0.07 mm, a first distance in the width directionof the groove measured from a first outer edge of the groove to a firstouter edge of the corresponding one of the first and second tuning forkarms being defined by W₁ and a second distance in the width direction ofthe groove measured from a second outer edge opposite the first outeredge of the groove to a second outer edge opposite the first outer edgeof the corresponding one of the first and second tuning fork arms beingdefined by W₃, and so that each of the first distance W₁ and the seconddistance W₃ is less than 0.015 mm; wherein an overall length of thequartz crystal resonator is less than 2.1 mm; wherein the length of thebase portion is less than 0.5 mm; wherein a first mounting arm isconnected to the base portion through a first connecting portion and asecond mounting arm is connected to the base portion through a secondconnecting portion, each of the first and second mounting arms extendingin a common direction with the first and second tuning fork arms outsidethe first and second tuning fork arms, and having a length within arange of 0.3 mm to 1.85 mm; wherein the case has first and secondmounting portions; wherein the first mounting arm is mounted on thefirst mounting portion of the case and the second mounting arm ismounted on the second mounting portion of the case; wherein the lid isconnected to the case to cover an open end of the case; wherein a firstelectrode is disposed on a surface of the groove formed the at least oneof the first and second main surfaces of the first vibrational portionof each of the first and second tuning fork arms; wherein a secondelectrode is disposed on a surface of each of the first and secondconnecting portions; wherein a third electrode is disposed on a surfaceof each of the first and second mounting arms; wherein a fourthelectrode is disposed on a surface of each of the first and secondmounting portions of the case; wherein the first electrode disposed onthe surface of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of one of thefirst and second tuning fork arms is connected to the third electrodedisposed on the surface of the first mounting arm through the secondelectrode disposed on the surface of the first connecting portion, andthe third electrode disposed on the surface of the first mounting arm isconnected to the fourth electrode disposed on the surface of the firstmounting portion of the case; and wherein the first electrode disposedon the surface of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of the other ofthe first and second tuning fork arms is connected to the thirdelectrode disposed on the surface of the second mounting arm through thesecond electrode disposed on the surface of the second connectingportion, and the third electrode disposed on the surface of the secondmounting arm is connected to the fourth electrode disposed on thesurface of the second mounting portion of the case.
 87. The quartzcrystal oscillator according to claim 86; wherein the overall length ofthe quartz crystal resonator is within a range of 1.02 mm to 1.95 mm;wherein the length of the base portion is within a range of 0.015 mm to0.49 mm; wherein the first width of the first vibrational portion ofeach of the first and second tuning fork arms is greater than 0.05 mmand less than 0.1 mm, the first vibrational portion of each of the firstand second tuning fork arms having a first length and the secondvibrational portion of each of the first and second tuning fork armshaving a second length; wherein the second length of second vibrationalportion of each of the first and second tuning fork arms is less thanthe first length of the first vibrational portion of the correspondingone of the first and second tuning fork arms; wherein the second widthof the second vibrational portion of each of the first and second tuningfork arms is greater than a spaced-apart distance between the secondvibrational portion of the first tuning fork arm and the secondvibrational portion of the second tuning fork arm; wherein the firstwidth of the first vibrational portion of each of the first and secondtuning fork arms is defined by W and a spaced-apart distance between thefirst vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm is defined by W₄;wherein the sum of the first width W of the first vibrational portion ofeach of the first and second tuning fork arms and the spaced-apartdistance W₄ between the first vibrational portion of the first tuningfork arm and the first vibrational portion of the second tuning fork armis defined by a width W₅=2W+W₄; wherein the width W₅ is less than awidth of the base portion; wherein the first connecting portion isconnected to the first mounting arm and has an outer edge; wherein oneof the first and second tuning fork arms adjoins the first mounting armand has an outer edge; wherein a distance in the length direction of thefirst connecting portion measured from the outer edge of the firstconnecting portion to the outer edge of one of the first and secondtuning fork arms is defined by L₄; and wherein the distance L₄ is withina range of 0.012 mm to 0.38 mm.
 88. The quartz crystal oscillatoraccording to claim 86; wherein the first vibrational portion of each ofthe first and second tuning fork arms has a first length and the secondvibrational portion of each of the first and second tuning fork arms hasa second width; wherein the second length of second vibrational portionof each of the first and second tuning fork arms is less than the firstlength of the first vibrational portion of the corresponding one of thefirst and second tuning fork arms; wherein the base portion has a firstbase portion, a second base portion and two cut portions formed betweenthe first and second base portions, each of the first and second tuningfork arms being connected to the first base portion of the base portion;wherein the first mounting arm is connected to the second base portionof the base portion through the first connecting portion and the secondmounting arm is connected to the second base portion of the base portionthrough the second connecting portion so that the second base portion ofthe base portion and the first and second connecting portions have aconcave shape or a U-shape with the first and second mounting arms;wherein the overall length of the quartz crystal resonator is within arange of 1.02 mm to 1.95 mm; wherein the length of the base portion iswithin a range of 0.015 mm to 0.49 mm; and wherein the second width ofthe second vibrational portion of each of the first and second tuningfork arms is greater than or equal to a spaced-apart distance betweenthe first vibrational portion of the first tuning fork arm and the firstvibrational portion of the second tuning fork arm.
 89. The quartzcrystal oscillator according to claim 86; wherein the length of the baseportion is within a range or 0.015 mm to 0.49 mm; wherein the baseportion has a first base portion including a first length, a second baseportion and a third base portion connecting the first base portion tothe second base portion, two cut portions being formed between the firstand second base portions, each of the first and second tuning fork armsbeing connected to the first base portion of the base portion; whereinthe first mounting arm is connected to the second base portion of thebase portion through the first connecting portion and the secondmounting arm is connected to the second base portion of the base portionthrough the second connecting portion so that the second base portion ofthe base portion and the first and second connecting portions have aconcave shape or a U-shape with the first and second mounting arms, thefirst vibrational portion of each of the first and second tuning forkarms having a first length and the second vibrational portion of each ofthe first and second tuning fork arms having a second length; andwherein the second length of second vibrational portion of each of thefirst and second tuning fork arms is less than the first length of thefirst vibrational portion of the corresponding one of the first andsecond tuning fork arms and greater than the first length of the firstbase portion of the base portion.
 90. The quartz crystal oscillatoraccording to claim 77; wherein each of the first and second tuning forkarms each having the first width and the second width greater than thefirst width has the opposite main surfaces including a first mainsurface and a second main surface, each of the first and second mainsurfaces of each of the first and second tuning fork arms having acentral linear portion; wherein the at least one groove comprises agroove formed in the central linear portion of at least one of the firstand second main surfaces of each of the first and second tuning forkarms so that a width of the groove formed in the central linear portionof the at least one of the first and second main surfaces of each of thefirst and second tuning fork arms is less than 0.07 mm, a first distancein the width direction of the groove measured from a first outer edge ofthe groove to a first outer edge of the corresponding one of the firstand second tuning fork arms being defined by W₁ and a second distance inthe width direction of the groove measured from a second outer edgeopposite the first outer edge of the groove to a second outer edgeopposite the first outer edge of the corresponding one of the first andsecond tuning fork arms being defined by W₃, and so that each of thefirst distance W₁ and the second distance W₃ is less than 0.015 mm, thewidth of the groove formed in the central linear portion of the at leastone of the first and second main surfaces of each of the first andsecond tuning fork arms being greater than or equal to each of the firstdistance W₁ and the second distance W₃; wherein an overall length of thequartz crystal resonator is within a range of 1.02 mm to 1.95 mm;wherein the length of the base portion is defined by L₁ and less than0.5 ruin and a width of the base portion is less than 0.55 mm; whereinat least one mounting arm is connected to the base portion through aconnecting portion so that the at least one mounting arm extends in acommon direction with the first and second tuning fork arms outside thefirst and second tuning fork arms and has a length within a range of 0.3mm to 1.85 mm, the at least one mounting arm having a first arm portionincluding a first width and a second arm portion including a secondwidth greater than the first width; wherein the case has a mountingportion; wherein the second arm portion of the at least one mounting armis mounted on the mounting portion of the case; wherein the lid isconnected to the case to cover an open end of the case; wherein anelectrode is disposed on each of a surface of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms, a surface ofthe connecting portion, a surface of the at least one mounting arm and asurface of the mounting portion of the case; and wherein the electrodedisposed on the surface of the groove formed in the central linearportion of the at least one of the first and second main surfaces of oneof the first and second tuning fork arms is connected to the electrodedisposed on the surface of the at least one mounting arm through theelectrode disposed on the surface of the connecting portion, and theelectrode disposed on the surface of the at least one mounting arm isconnected to the electrode disposed on the surface of the mountingportion of the case.
 91. The quartz crystal oscillator according toclaim 90; wherein the length L₁ of the base portion is within a range of0.015 mm to 0.49 mm and the width of the base portion is within a rangeof 0.15 mm to 0.53 mm; wherein the width of the at least one mountingarm is within a range of 0.08 mm to 0.4 mm and greater than the width ofeach of the first and second tuning fork arms; wherein the connectingportion has a length L₂; wherein the length L₁ of the base portion isgreater than the length L₂ of the connecting portion; wherein the widthof the at least one mounting arm is defined by W₆; wherein the firstwidth of each of the first and second tuning fork arms is defined by W;wherein a ratio of the width W₆ of the at least one mounting arm and thefirst width W of each of the first and second tuning fork arms isdefined by W₆₁=W₆/W; wherein the ratio W₆₁ is within a range of 1.1 to3.8; wherein the first width of each of the first and second tuning forkarms is defined by W and a spaced-apart distance between the firsttuning fork arm and the second tuning fork arm is defined by W₄; whereinthe sum of the first width W of each of the first and second tuning forkarms and the spaced-apart distance W₄ between the first tuning fork armand the second tuning fork arm is defined by a width W₅=2W+W₄; whereinthe width W₅ is less than the width of the base portion, the connectingportion being connected to the at least one mounting arm and having anouter edge, one of the first and second tuning fork arms adjoining theat least one mounting arm and having an outer edge; wherein a distancein the length direction of the connecting portion measured from theouter edge of the connecting portion to the outer edge of one of thefirst and second tuning fork arms is defined by L₄; and wherein thedistance L₄ is within a range of 0.012 mm to 0.38 mm.
 92. The quartzcrystal oscillator according to claim 90; wherein the base portion has afirst base portion, a second base portion and a third base portionconnecting the first base portion to the second base portion, two cutportions being formed between the first and second base portions, eachof the first and second tuning fork arms being connected to the firstbase portion of the base portion; wherein the at least one mounting armhas first and second mounting arms; wherein the first mounting arm isconnected to the second base onion of the base portion through a firstconnecting portion and the second mounting arm is connected to thesecond base portion of the base portion through a second connectingportion so that the second base portion of the base portion and thefirst and second connecting portions have a concave shape or a U-shapewith the first and second mounting arms; wherein the case has first andsecond mounting portions; wherein the first mounting arm is mounted onthe first mounting portion of the case and the second mounting arm ismounted on the second mounting portion of the case, each of the firstand second connecting portions having a width W_(S) and a length L₂;wherein the width W_(S) of each of the first and second connectingportions is within a range of 0.015 mm to 0.14 mm and less than thelength L₂ of each of the first and second connecting portions, each ofthe first and second mounting arms having a width and a length; whereinthe width of each of the first and second mounting arms is within arange of 0.025 mm to 0.4 mm and the length of each of the first andsecond mounting arms is within a range of 0.3 mm to 1.85 mm; wherein afirst electrode is disposed on a surface of the groove formed in thecentral linear portion of the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms; wherein asecond electrode is disposed on a surface of each of the first andsecond connecting portions; wherein a third electrode is disposed on asurface of each of the first and second mounting arms; wherein a fourthelectrode is disposed on a surface of each of the first and secondmounting portions of the case; wherein the first electrode disposed onthe surface of the groove formed in the central linear portion of the atleast one of the first and second main surfaces of one of the first andsecond tuning fork arms is connected to the third electrode disposed onthe surface of the first mounting arm through the second electrodedisposed on the surface of the first connecting portion, and the thirdelectrode disposed on the surface of the first mounting arm is connectedto the fourth electrode disposed on the surface of the first mountingportion of the case; wherein the first electrode disposed on the surfaceof the groove formed in the central linear portion of the at least oneof the first and second main surfaces of the other of the first andsecond tuning fork arms is connected to the third electrode disposed onthe surface of the second mounting arm through the second electrodedisposed on the surface of the second connecting portion, and the thirdelectrode disposed on the surface of the second mounting arm isconnected to the fourth electrode disposed on the surface of the secondmounting portion of the case; wherein the width of the groove formed inthe central linear portion of the at least one of the first and secondmain surfaces of each of the first and second tuning fork arms is withina range of 0.038 mm to 0.05 mm; wherein each of the first distance W₁and the second distance W₃ is within a range of 0.001 mm to 0.012 mm,the first connecting portion being connected to the first mounting armand having an outer edge, one of the first second tuning fork armsadjoining the first mounting arm and having an outer edge, the secondconnecting portion being connected to the second mounting arm and havingan outer edge, and the other of the first and second tuning fork armsadjoining the second mounting arm and having an outer edge; wherein afirst distance length direction of the first connecting portion measuredfrom the outer edge of the first connecting portion to the outer edge ofone of the first and second tuning fork arms is defined by L₄; wherein asecond distance in the length direction of the second connecting portionmeasured from the outer edge of the second connecting portion to theouter edge of the other of the first and second tuning fork arms isdefined by L₄; and wherein each of the first distance L₄ and the seconddistance L₄ is within a range of 0.012 mm to 0.38 mm.
 93. A quartzcrystal resonator comprising: a base portion; and first and secondtuning fork arms connected to the base portion, the base portion havinga length greater than a spaced-apart distance between the first tuningfork arm and the second tuning fork arm, each of the first and secondtuning fork arms having a width less than 0.18 mm and opposite mainsurfaces, at least one groove being formed in at least one of theopposite main surfaces of each of the first and second tuning fork armsso that a length of the at least one groove formed in the at least oneof the opposite main surfaces of each of the first and second tuningfork arms is within a range of 0.45 mm to 1.25 mm, and an electrodebeing disposed on a surface of the at least one groove formed in the atleast one of the opposite main surfaces of each of the first and secondtuning fork arms so that the electrode disposed on the surface of the atleast one groove formed in the at least one of the opposite mainsurfaces of the first tuning fork arm has an electrical polarityopposite to an electrical polarity of the electrode disposed on thesurface of the at least one groove formed in the at least one of theopposite main surfaces of the second tuning fork arm.
 94. The quartzcrystal resonator according to claim 93; wherein each of the first andsecond tuning fork arms comprises a first vibrational portion having afirst width, and a second vibrational portion having a second widthgreater than the first width, the first vibrational portion of each ofthe first and second tuning fork arms having the opposite main surfacesincluding a first main surface and a second main surface; wherein the atleast one groove comprises a groove having a width and formed in atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms, a firstdistance in the width direction of the groove measured from a firstouter edge of the groove to a first outer edge of the corresponding oneof the first and second tuning fork arms being defined by W₁ and asecond distance in the width direction of the groove measured from asecond outer edge opposite the first outer edge of the groove to asecond outer edge opposite the first outer edge of the corresponding oneof the first and second tuning fork arms being defined by W₃; whereinthe width of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms is greater than each of the firstdistance W₁ and the second distance W₃ and less than 0.07 mm; whereineach of the first distance WI and the second distance W₃ is less than0.015 mm; and wherein an overall length of the quartz crystal resonatoris less than 2.1 mm.
 95. The quartz crystal resonator according to claim94; wherein the overall length of the quartz crystal resonator is withina range of 1.02 mm to 1.95 mm; wherein the first vibrational portion ofeach of the first and second tuning fork arms is formed between thesecond vibrational portion of the corresponding one of the first andsecond tuning fork arms and the base portion so that a length of thefirst vibrational portion is greater than a length of the secondvibrational portion; and wherein the base portion has the length lessthan 0.5 mm and a plurality of different widths.
 96. The quartz crystalresonator according to claim 94; wherein the first vibrational portionof each of the first and second tuning fork arms has a first length;wherein the second vibrational portion of each the first and secondtuning fork arms has a second length greater than the second width andless than the first length: wherein the base portion has a first baseportion including a first width and a second base portion including asecond width different from the first width; wherein each of the firstand second tuning fork arms is directly connected to the first baseportion of the base portion; and wherein the base portion has the lengthwithin a range of 0.015 mm to 0.49 mm.
 97. The quartz crystal resonatoraccording to claim 94; wherein a width of the base portion is less than0.55 mm and the length of the base portion is within a range of 0.015 mmto 0.49 mm: and wherein at least one mounting arm has a length within arange of 0.3 mm to 1.85 mm and is connected to the base portion througha connecting portion having a width Ws less than 0.41 mm so that the atleast one mounting arm extends in a common direction with the first andsecond tuning fork arms outside the first and second tuning fork arms.98. The quartz crystal resonator according to claim 94; wherein thewidth of each of the first and second tuning fork arms is greater than0.05 mm and less than 0.1 mm, and a length of each of the first andsecond tuning fork arms is within a range of 0.53 mm to 1.46 mm; whereinthe first vibrational portion of each of the first and second tuningfork arms has a first length; wherein the second vibrational portion ofeach the first and second tuning fork arms has a second length greaterthan the second width and less than the first length; wherein the baseportion has a first base portion, a second base portion and two cutportions formed between the first and second base portions, each of thefirst and second timing fork arms being connected to the first baseportion of the base portion; wherein two mounting arms having first andsecond mounting arms each including a length within a range of 0.3 mm to1.85 mm are connected to the second base portion of the base portion;and wherein the first mounting arm is connected to the second baseportion of the base portion through a first connecting portion and thesecond mounting arm is connected to the second base portion of the baseportion through a second connecting portion so that the second baseportion of the base portion and the first and second connecting portionshave a concave shape with the first and second mounting arms.
 99. Thequartz crystal resonator according to claim 94; wherein a length of eachof the first and second tuning fork arms is within a range of 0.53 mm to1.46 mm; wherein the base portion has the length less than 0.5 mm, and afirst base portion, a second base portion and two cut portions formedbetween the first and second base portions, each of the first and secondtuning fork arms being connected to the first base portion of the baseportion; wherein a first mounting arm is connected to the second baseportion of the base portion through a first connecting portion and asecond mounting arm is connected to the second base portion of the baseportion through a second connecting portion so that each of the firstand second mounting arms extends in a common direction with the firstand second tuning fork arms outside the first and second tuning forkarms, each of the first and second mounting arms having a width; andwherein the first width of the first vibrational portion of each of thefirst and second tuning fork arms is greater than the width of each ofthe first and second mounting arms.
 100. The quartz crystal resonatoraccording to claim 93; wherein each of the first and second tuning forkarms comprises a plurality of different widths having a first width anda second width different from the first with, the opposite main surfacesof each of the first and second tuning fork arms having a first mainsurface and a second main surface; wherein the at least one groovecomprises a groove having a width and formed in at least one of thefirst and second main surfaces of each of the first and second tuningfork arms, a first distance in the width direction of the groovemeasured from a first outer edge of the groove to a first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W3; wherein the width of the groove formed in the at least one of thefirst and second main surfaces of each of the first and second tuningfork arms is greater than each of the first distance Wi and the seconddistance W₃ and less than 0.07 mm ; wherein each of the first distanceW₁ and the second distance W₃ is less than 0.015 mm; wherein an overalllength of the quartz crystal resonator is less than 2.1 mm; and whereinthe base portion has the length less than 0.5 mm.
 101. The quartzcrystal resonator according to claim 100; wherein the overall length ofthe quartz crystal resonator is within a range of 1.02 mm to 1.95 mm;wherein the length of the base portion is within a range of 0.015 mm to0.49 mm; wherein the base portion has a first base portion, a secondbase portion and two cut portions formed between the first and secondbase portions, each of the first and second tuning fork arms beingconnected to the first base portion of the base portion; wherein twomounting arms having first and second mounting arms are connected to thesecond base portion of the base portion, each of the first and secondmounting arms having a length within a range of 0.3 mm to 1.85 mm, and afirst arm portion including a first width and a second arm portionincluding a second width greater than the first width; and wherein thefirst arm portion of the first mounting arm is connected to the secondbase portion of the base portion through a first connecting portion andthe first arm portion of the second mounting arm is connected to thesecond base portion of the base portion through a second connectingportion so that the second base portion of the base portion and thefirst and second connecting portions have a concave shape with the firstand second arm portions of the first and second mounting arms.
 102. Thequartz crystal resonator according to claim 100; wherein the overalllength of the quartz crystal resonator is within a range of 1.02 mm to1.95 mm; wherein the length of the base portion is within a range of0.015 mm to 0.49 mm; wherein two mounting arms having first and secondmounting arms are connected to the base portion, each of the first andsecond mounting arms having a length within a range of 0.3 mm to 1.85mm, and a first arm portion including a first width and a second armportion including a second width greater than the first width; andwherein the first arm portion of the first mounting arm is connected tothe base portion through a first connecting portion and the first armportion of the second mounting arm is connected to the base portionthrough a second connecting portion.
 103. The quartz crystal resonatoraccording to claim 93; wherein the opposite main surfaces of each of thefirst and second tuning fork arms have a first main surface and a secondmain surface; wherein the at least one groove comprises a groove formedin at least one of the first and second main surfaces of each of thefirst and second tuning fork arms so that a width of the grooveformedned in the at least one of the first and second main surfaces ofeach of the first and second tuning fork arms is greater than each of afirst distance W₁ in the width direction of the groove measured from afirst outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms and a seconddistance W₃ in the width direction of the groove measured from a secondouter edge opposite the first outer edge of the groove to a second outeredge opposite the first outer edge of the corresponding one of the firstand second tuning fork arms; wherein a first electrode is disposed on asurface of the groove formed in the at least one of the first and secondmain surfaces of each of the first and second tuning fork arms and asecond electrode is disposed on a side surface of each of the first andsecond tuning fork arms; wherein the first electrode disposed on thesurface of the groove formed in the at least one of the first and secondmain surfaces of the first tuning fork arm is connected to the secondelectrode disposed on the side surface of the second tuning fork arm.and the first electrode disposed on the surface of the groove formed inthe at least one of the first and second main surfaces of the secondtuning fork arm is connected to the second electrode disposed on theside surface of the first tuning fork arm; wherein the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the first tuning fork arm has anelectrical polarity opposite to an electrical polarity of the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the second tuning fork arm;and further comprising a quality factor Q₁ and a capacitance ratio r₁ ofa fundamental mode of vibration, and a quality factor Q₂ and acapacitance ratio r₂ of a second overtone mode of vibration, a firstmerit value M₁ being defined by M₁=Q₁/r₁, a second merit value M₂ beingdefined by M₂ =W₂/r₂; wherein the first merit value M₁ is greater thanthe second merit value M₂ and the second merit value M₂ is less than 30;wherein the base portion has a first base portion and a second baseportion, at least one cut portion being formed between the first andsecond base portions, each of the first and second tuning fork armsbeing connected to the first base portion of the base portion; andproviding a case having a mounting portion and a lid; wherein the secondbase portion of the base portion is mounted on the mounting portion ofthe case; and wherein the lid is connected to the case to cover an openend of the case.
 104. The quartz crystal resonator according to claim93; wherein each of the first and second tuning fork arms comprises afirst vibrational portion having a first width W and a first length, anda second vibrational portion having a second width greater than thefirst width and a second length greater than the second width and lessthan the first length, the first vibrational portion of each of thefirst and second tuning fork arms having the opposite main surfacesincluding a first main surface and a second main surface; wherein the atleast one groove comprises a groove formed in at least one of the firstand second main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms, a first distance in the widthdirection of the groove measured from a first outer edge of the grooveto a first outer edge of the corresponding one of the first and secondtuning fork arms being defined by W₁ and a second distance in the widthdirection of the groove measured from a second outer edge opposite thefirst outer edge of the groove to a second outer edge opposite the firstouter edge of the corresponding one of the first and second tuning forkarms being defined by W₃; wherein a width W₂ of the groove formed in theat least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms isgreater than each of the first distance W₁ and the second distance W₃:wherein a first electrode is disposed on a surface of the groove formedin the at least one of the first and second main surfaces of the firstvibrational portion of each of the first and second tuning fork arms anda second electrode is disposed on a side surface of the firstvibrational portion of each of the first and second tuning fork arms;wherein the first electrode disposed on the surface of the groove formedin the at least one of the first and second main surfaces of the firstvibrational portion of the first tuning fork arm is connected to thesecond electrode disposed on the side surface of the first vibrationalportion of the second tuning fork arm, and the first electrode disposedon the surface of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of the secondtuning fork arm is connected to the second electrode disposed on theside surface of the first vibrational portion of the first tuning forkarm; wherein the first electrode disposed on the surface of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of the first tuning fork arm has an electricalpolarity opposite to an electrical polarity of the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the first vibrational portion of thesecond tuning fork arm; and further comprising a quality factor Q₁ and acapacitance ratio r₁ of a fundamental mode of vibration, and a qualityfactor Q₂ and a capacitance ratio r₂ of a second overtone mode ofvibration, a first merit value M₁ being defined by M₁=Q₁/r₁, a secondmerit value M₂ being defined by M₂=Q₂/r₂; wherein the first merit valueM₁ is greater than the second merit value M₂; and wherein the secondmerit value M₂ is less than
 30. 105. The quartz crystal resonatoraccording to claim 104; wherein a ratio of the width W₂ of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms and the first width W of the first vibrational portion of thecorresponding one of the first and second tuning fork arms is defined byW₂₁=W₂/W; and wherein the ratio W₂₁ is within a range of 0.35 to 0.95.106. A quartz crystal resonator comprising: a base portion; and firstand second tuning fork arms connected to the base portion, the baseportion having a length greater than a spaced-apart distance between thefirst tuning fork arm and the second tuning fork arm, each of the firstand second tuning fork arms having a plurality of different widthsincluding a first width and a second width greater than the first width,and opposite main surfaces, at least one groove being formed in at leastone of the opposite main surfaces of each of the first and second tuningfork arms so that a length of the at least one groove formed in the atleast one of the opposite main surfaces of each of the first and secondtuning fork arms is within a range of 0.3 mm to 0.79 mm, and anelectrode being disposed on a surface of the at least one groove formedin the at least one of the opposite main surfaces of each of the firstand second tuning fork arms so that the electrode disposed on thesurface of the at least one groove formed in the at least one of theopposite main surfaces of the first tuning fork arm has an electricalpolarity opposite to an electrical polarity of the electrode disposed onthe surface of the at least one groove formed in the at least one of theopposite main surfaces of the second tuning fork arm.
 107. The quartzcrystal resonator according to claim 106: wherein the opposite mainsurfaces of each of the first and second tuning fork arms have a firstmain surface and a second main surface; wherein the at least one groovecomprises a groove formed in at least one of the first and second mainsurfaces of each of the first and second tuning fork arms so that awidth of the groove formed in the at least one of the first and secondmain surfaces of each of the first and second tuning fork arms is lessthan 0.07 mm and greater than each of a first distance W₁ in the widthdirection of the groove measured from a first outer edge of the grooveto a first outer edge of the corresponding one of the first and secondtuning fork arms and a second distance W₃ in the width direction of thegroove measured from a second outer edge opposite the first outer edgeof the groove to a second outer edge opposite the first outer edge ofthe corresponding one of the first and second tuning fork arms. and sothat each of the first distance W₁ and the second distance W₃ is lessthan 0.015 mm; wherein an overall length of the quartz crystal resonatoris less than 2.1 mm and a width of the base portion is less than 0.55mm; and wherein the base portion has the length less than 0.5mm. 108.The quartz crystal resonator according to claim 107: wherein the overalllength of the quartz crystal resonator is within a range of 1.02 mm to1.95 mm; wherein the width of the base portion is within a range of 0.15mm to 0.53 mm; wherein a first electrode is disposed on a surface of thegroove formed in the at least one of the first and second main surfacesof each of the first and second tuning fork arms and a second electrodeis disposed on a side surface of each of the first and second tuningfork arms; wherein the first electrode disposed on the surface of thegroove formed in the at least one of the first and second main surfacesof the first tuning fork arm is connected to the second electrodedisposed on the side surface of the second tuning fork arm. and thefirst electrode disposed on the surface of the groove formed in the atleast one of the first and second main surfaces of the second tuningfork arm is connected to the second electrode disposed on the sidesurface of the first tuning fork arm; wherein the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the first tuning fork arm has anelectrical polarity opposite to an electrical polarity of the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the second tuning fork arm;and further comprising a series resistance R₁, a quality factor Q₁ and acapacitance ratio r₁ of a fundamental mode of vibration, and a seriesresistance R₂, a quality factor Q₂ and a capacitance ratio r₂ of asecond overtone mode of vibration, a first merit value M₁ being definedby M₁=Q₁/r₁, a second merit value M₂ being defined by M₂=Q₂/r₂; whereinthe first merit value M₁ is greater than the second merit value M₂;wherein the second merit value M₂ is less than 30; wherein the seriesresistance R₁ of the fundamental mode of vibration is less than theseries resistance R₂ of the second overtone mode of vibration: andwherein the capacitance ratio r₂ of the second overtone mode ofvibration is greater than
 1500. 109. The quartz crystal resonatoraccording to claim 107: wherein the length of the base portion is withina range of 0.015 mm to 0.49 mm; wherein a length of each of the firstand second tuning fork arms is within a range of 0.56 mm to 1.15 mm;wherein each of the first and second tuning fork arms comprises a firstvibrational portion having the first width and a first length. and asecond vibrational portion having the second width greater than thefirst width and a second length less than the first length; and whereinthe second width of the second vibrational portion of each of the firstand second tuning fork arms is less than the second length of the secondvibrational portion of the corresponding one of the first and secondtuning fork arms.
 110. The quartz crystal resonator according to claim107: wherein each of the first and second tuning fork arms has a widthless than 0.1 mm and a length less than 1.56 mm; wherein the baseportion has a first base portion, a second base portion and two cutportions formed between the first and second base portions, each of thefirst and second tuning fork arms being connected to the first baseportion of the base portion; wherein two mounting arms having first andsecond mounting arms are connected to the second base portion of thebase portion; wherein the first mounting arm is connected to the secondbase portion of the base portion through a first connecting portion andthe second mounting arm is connected to the second base portion of thebase portion through a second connecting portion so that the second baseportion of the base portion and the first and second connecting portionshave a concave shape with the first and second mounting arms. each ofthe first and second mounting arms having a width and a length within arange of 0.3 mm to 1.85 mm; and wherein the width of each of the firstand second mounting arms is greater than the width of each of the firstand second tuning fork arms.
 111. The quartz crystal unit comprising: acase; a lid; and a quartz crystal resonator having a base portion, andfirst and second tuning fork arms connected to the base portion, thebase portion having a length greater than a spaced-apart distancebetween the first tuning fork arm and the second tuning fork arm, eachof the first and second tuning fork arms having a width less than 0.18mm and opposite main surfaces, at least one groove being formed in atleast one of the opposite main surfaces of each of the first and secondtuning fork arms, an electrode being disposed on a surface of the atleast one groove formed in the at least one of the opposite mainsurfaces of each of the first and second tuning fork arms so that theelectrode disposed on the surface of the at least one groove formed inthe at least one of the opposite main surfaces of the first tuning forkarm has an electrical polarity opposite to an electrical polarity of theelectrode disposed on the surface of the at least one groove formed inthe at least one of the opposite main surfaces of the second tuning forkarm, the quartz crystal resonator having a quality factor Q₁ and acapacitance ratio r₁ of a fundamental mode of vibration, and a qualityfactor Q₂ and a capacitance ratio r₂ of a second overtone mode ofvibration, a first merit value M₁ being defined by M₁=Q₁/r₁, a secondmerit value M₂ being defined by M₂=Q₂/r₂, and the first merit value M₁being greater than the second merit value M₂.
 112. the quartz crystalunit according to claim 111; wherein each of the first and second tuningfork arms comprises a first vibrational portion having a first width,and a second vibrational portion having a second width greater than thefirst width, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface; wherein the at least onegroove comprises a groove formed in at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms, the first vibrational portion of each of thefirst and second tuning fork arms having a first length and the secondvibrational portion of each of the first and second tuning fork armshaving a second length greater than the second width; wherein the secondlength of the second vibrational portion of each of the first and secondtuning fork arms is less than the first length of the first vibrationalportion of the corresponding one of the first and second tuning forkarms; wherein a first electrode is disposed on a surface of the grooveformed in the at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms, and a second electrode is disposed on a side surface of the firstvibrational portion of each of the first and second tuning fork arms;wherein each of the first electrode disposed on the surface of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms, and the second electrode disposed on the side surface of thefirst vibrational portion of each of the first and second tuning forkarms is formed so that the second merit value M₂ is less than 30;wherein the base portion has a first base portion, a second base portionand a cut portion formed between the first and second base portions,each of the first and second tuning fork arms being connected to thefirst base portion of the base portion. the case having a mountingportion; and wherein the second base portion of the base portion ismounted on the mounting portion of the case.
 113. The quartz crystalunit according to claim 111; wherein each of the first and second tuningfork arms comprises a first vibrational portion having a first width anda first length, and a second vibrational portion having a second widthgreater than the first width and a second length less than the firstlength, the first vibrational portion of each of the first and secondtuning fork arms having the opposite main surfaces including a firstmain surface and a second main surface; wherein the at least one groovecomprises a groove formed in at least one of the first and second mainsurfaces of the first vibrational portion of each of the first andsecond tuning fork arms so that a length of the groove formed in the atleast one of the first and second main surfaces of the first vibrationalportion of each of the first and second tuning fork arms is within arange of 0.45 mm to 1.25 mm; wherein a first electrode is disposed on asurface of the groove formed in the at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms, and a second electrode is disposed on a sidesurface of the first vibrational portion of each of the first and secondtuning fork arms so that the capacitance ratio r₂ of the second overtonemode of vibration is greater than 1500, the case having a mountingportion; wherein the base portion is mounted on the mounting portion ofthe case; and wherein the lid is connected to the case to cover an openend of the case.
 114. The quartz crystal unit according to claim 111;wherein the opposite main surfaces of each of the first and secondtuning fork arms have a first main surface and a second main surface;wherein the at least one groove comprises a groove formed in at leastone of the first and second main surfaces of each of the first andsecond tuning fork arms so that a width of the groove formed in the atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms is greater than each of a first distance W₁ inthe width direction of the groove measured from a first outer edge ofthe groove to a first outer edge of the corresponding one of the firstand second tuning fork arms and a second distance W₃ in the widthdirection of the groove measured from a second outer edge opposite thefirst outer edge of the groove to a second outer edge opposite the firstouter edge of the corresponding one of the first and second tuning forkarms; wherein a first electrode is disposed on a surface of the grooveformed in the at least one of the first and second main surfaces of eachof the first and second tuning fork arms and a second electrode isdisposed on a side surface of each of the first and second tuning forkarms; wherein the first electrode disposed on the surface of the grooveformed in the at least one of the first and second main surfaces of thefirst tuning fork arm is connected to the second electrode disposed onthe side surface of the second tuning fork arm. and the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the second tuning fork arm isconnected to the second electrode disposed on the side surface of thefirst tuning fork arm so that the first electrode disposed on thesurface of the groove formed in the at least one of the first and secondmain surfaces of the first tuning fork arm has an electrical polarityopposite to an electrical polarity of the first electrode disposed onthe surface of the groove formed in the at least one of the first andsecond main surfaces of the second tuning fork arm, and so that thesecond merit value M₂ is less than 30; wherein the base portion has afirst base portion and a second base portion, at least one cut portionbeing formed between the first and second base portions, each of thefirst and second tuning fork arms being connected to the first baseportion of the base portion, the case having a mounting portion; whereinthe second base portion of the base portion is mounted on the mountingportion of the case; and wherein the lid is connected to the case tocover an open end of the case.
 115. The quartz crystal unit according toclaim 111; wherein the opposite main surfaces of each of the first andsecond tuning fork arms have a first main surface and a second mainsurface; wherein the at least one groove comprises a groove formed in atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms, a first distance in the width direction of thegroove measured from a first outer edge of the groove to a first outeredge of the corresponding one of the first and second tuning fork armsbeing defined by W₁, a second distance in the width direction of thegroove measured from a second outer edge opposite the first outer edgeof the groove to a second outer edge opposite the first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₃; wherein a width of the groove formed in the at least oneof the first and second main surfaces of each of the first and secondtuning fork arms is greater than each of the first distance W₁ and thesecond distance W₃; wherein a first electrode is disposed on a surfaceof the groove formed in the at least one of the first and second mainsurfaces of each of the first and second tuning fork arms and a secondelectrode is disposed on a side surface of each of the first and secondtuning fork arms; wherein the first electrode disposed on the surface ofthe groove formed in the at least one of the first and second mainsurfaces of the first tuning fork arm is connected to the secondelectrode disposed on the side surface of the second tuning fork arm,and the first electrode disposed on the surface of the groove formed inthe at least one of the first and second main surfaces of the secondtuning fork arm is connected to the second electrode disposed on theside surface of the first tuning fork arm so that the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the first tuning fork arm has anelectrical polarity opposite to an electrical polarity of the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the second tuning fork arm,and so that the first merit value M₁ is greater than 65; wherein thebase portion has a plurality of different widths including a first widthand a second width greater than the first width, the case having amounting portion; wherein the base portion is mounted on the mountingportion of the case; and wherein the lid is connected to the case tocover an open end of the case.
 116. The quartz crystal unit according toclaim 111; wherein each of the first and second tuning fork armscomprises a plurality of different widths having a first, width and asecond width different from the first with. the opposite main surfacesof each of the first and second tuning fork arms having a first mainsurface and a second main surface; wherein the at least one groovecomprises a groove formed in at least one of the first and second mainsurfaces of each of the first and second tuning fork arms, a firstdistance in the width direction of the groove measured from a firstouter edge of the groove to a first outer edge of the corresponding oneof the first and second tuning fork arms being defined by W₁ and asecond distance in the width direction of the groove measured from asecond outer edge opposite the first outer edge of the groove to asecond outer edge opposite the first outer edge of the corresponding oneof the first and second tuning fork arms being defined by W₃; wherein awidth of the groove formed in the at least one of the first and secondmain surfaces of each of the first and second tuning fork arms isgreater than each of the first distance W₁ and the second distance W₃and less than 0.07 mm; wherein each of the first distance W₁ and thesecond distance W₃ is less than 0.015 mm; wherein an overall length ofthe quartz crystal resonator is less than 2.1 mm; and wherein the baseportion has the length less than 0.5 mm.
 117. The quartz crystal unitaccording to claim 116; wherein each of the first and second tuning forkarms comprises a first vibrational portion having the first width, and asecond vibrational portion having the second width different from thefirst width: wherein the first width of the first vibrational portion ofeach of the first and second tuning fork arms is defined by W; wherein aspaced-apart distance between the first vibrational portion of the firsttuning fork arm and the first vibrational portion of the second tuningfork arm is defined by W₄; wherein the sum of the width W of the firstvibrational portion of each of the first and second tuning fork arms andthe spaced-apart distance W4 between the first vibrational portion ofthe first tuning fork arm and the first vibrational portion of thesecond tuning fork arm is defined by a width W₅ =2W +W₄; wherein thewidth W₅ is within a range of 0.15 mm to 0.52 mm; and wherein theoverall length of the quartz crystal resonator is within a range of 0.8mm to 1.95 mm and the length of the base portion is within a range of0.015 mm to 0.49 mm.
 118. The quartz crystal unit according to claim116; wherein the overall length of the quartz crystal resonator iswithin a range of 1.02 mm to 1.95 mm; wherein the length of the baseportion is within a range of 0.015 mm to 0.49 mm; wherein at least onemounting arm having a first mounting arm portion including a first widthand a second mounting arm portion including a second width greater thanthe first width is connected to the base portion through a connectingportion so that the at least one mounting arm extends in a commondirection with the first and second tuning fork arms outside the firstand second tuning fork arms and has a length within a range of 0.3 mm to1.85 mm; wherein the case has a mounting portion; wherein the secondmounting arm portion of the at least one mounting arm is mounted on themounting portion of the case; and wherein the lid is connected to thecase to cover an open end of the case.
 119. The quartz crystal unitaccording to claim 116; wherein the length of the base portion is withina range of 0.015 mm to 0.49 mm; wherein each of the first and secondtuning fork arms comprises a first vibrational portion having the firstwidth and a first length, and a second vibrational portion having thesecond width greater than the first width and a second length greaterthan the second width and less than the first length, the firstvibrational portion of each of the first and second tuning fork armshaving the opposite main surfaces including a third main surface and afourth main surface; wherein the at least one groove comprises a grooveformed in at least one of the third and fourth main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms so that a width of the groove formed in the at least one of thethird and fourth main surfaces of the first vibrational portion of eachof the first and second tuning fork arms is greater than each of a thirddistance in the width direction of the groove measured from a thirdouter edge of the groove to a third outer edge of the corresponding oneof the first and second tuning fork arms and a fourth distance in thewidth direction of the groove measured from a fourth outer edge oppositethe third outer edge of the groove to a fourth outer edge opposite thethird outer edge of the corresponding one of the first and second tuningfork arms and less than 0.07 mm; wherein each of the third distance andthe fourth distance is less than 0.015 mm; wherein the base portion hasa first base portion, a second base portion and two cut portions formedbetween the first and second base portions, each of the first and secondtuning fork arms being connected to the first base portion of the baseportion; wherein a first mounting arm is connected to the second baseportion of the base portion through a first connecting portion and asecond mounting arm is connected to the second base portion of the baseportion through a second connecting portion so that each of the firstand second mounting arms extends in a common direction with the firstand second tuning fork arms outside the first and second tuning forkarms, each of the first and second mounting arms having a width; andwherein the first width of the first vibrational portion of each of thefirst and second tuning fork arms is greater than the width of each ofthe first and second mounting arms.
 120. The quartz crystal oscillatorcomprising: a quartz crystal oscillating circuit having a plurality ofcapacitors, a resistor and a quartz crystal unit comprised of a case, alid and a quartz crystal resonator having a base portion, and first andsecond tuning fork arms connected to the base portion, the base portionhaving a length greater than a spaced-apart distance between the firsttuning fork arm and the second tuning fork arm. each of the first andsecond tuning fork arms having a width less than 0.18 mm and oppositemain surfaces, at least one groove being formed in at least one of theopposite main surfaces of each of the first and second tuning fork arms,an electrode being disposed on a surface of the at least one grooveformed in the at least one of the opposite main surfaces of each of thefirst and second tuning fork arms so that the electrode disposed on thesurface of the at least one groove formed in the at least one of theopposite main surfaces of the first tuning fork arm has an electricalpolarity opposite to an electrical polarity of the electrode disposed onthe surface of the at least one groove formed in the at least one of theopposite main surfaces of the second tuning fork arm, the quartz crystalresonator having a quality factor Q₁ and a capacitance ratio r₁ of afundamental mode of vibration, and a quality factor Q₂ and a capacitanceratio r₂ of a second overtone mode of vibration, a first merit value M₁being defined by M₁=Q₁/r₁, a second merit value M₂ being defined byM₂=Q₂/r₂, and the first merit value M₁ being greater than the secondmerit value M₂.
 121. The quartz crystal oscillator according to claim120; wherein each of the first and second tuning fork arms comprises aplurality of different widths having a first width and a second widthdifferent from the first with, the opposite main surfaces of each of thefirst and second tuning fork arms having a first main surface and asecond main surface; wherein the at least one groove comprises a grooveformed in at least one of the first and second main surfaces of each ofthe first and second tuning fork arms, a first distance in the widthdirection of the groove measured from a first outer edge of the grooveto a first outer edge of the corresponding one of the first and secondtuning fork arms being defined by W₁ and a second distance in the widthdirection of the groove measured from a second outer edge opposite thefirst outer edge of the groove to a second outer edge opposite the firstouter edge of the corresponding one of the first and second tuning forkarms being defined by W₃; wherein a width of the groove formed in the atleast one of the first and second main surfaces of each of the first andsecond tuning fork arms is greater than each of the first distance W₁and the second distance W₃ and less than 0.07 mm: wherein each of thefirst distance W1 and the second distance W₃ is less than 0.015 mm;wherein an overall length of the quartz crystal resonator is less than2.1 mm; and wherein the base portion has the length less than 0.5 mm.122. The quartz crystal oscillator according to claim 121; wherein thelength of the base portion is within a range of 0.015 mm to 0.49 mm;wherein the base portion has a first base portion, a second base portionand a cut portion formed between the first and second base portions,each of the first and second tuning fork arms being connected to thefirst base portion of the base portion; wherein two mounting arms havingfirst and second mounting arms are connected to the second base portionof the base portion, each of the first and second mounting arms having alength within a range of 0.3 mm to 1.85 mm; wherein the first mountingarm is connected to the second base portion of the base portion througha first connecting portion and the second mounting arm is connected tothe second base portion of the base portion through a second connectingportion, each of the first and second mounting arms having a width andextending in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms; wherein the width ofeach of the first and second mounting arms is greater than the firstwidth of each of the first and second tuning fork arms; wherein the casehas first and second mounting portions; wherein the first mounting armis mounted on the first mounting portion of the case and the secondmounting arm is mounted on the second mounting portion of the case; andwherein the lid is connected to the case to cover an open end of thecase.
 123. The quartz crystal oscillator according to claim 121; whereinthe overall length of the quartz crystal resonator is within a range of1.02 mm to 1.95 mm; wherein the length of the base portion is within arange of 0.015 mm to 0.49 mm; wherein the base portion has a first baseportion, a second base portion and a third base portion connecting thefirst base portion to the second base portion, each of the first andsecond tuning fork arms being connected to the first base portion of thebase portion; wherein two mounting arms having first and second mountingarms are connected to the second base portion of the base portion, eachof the first and second mounting arms having a length within a range of0.3 mm to 1.85 mm, and a first arm portion including a first width and asecond arm portion including a second width greater than the firstwidth; wherein the first arm portion of the first mounting arm isconnected to the second base portion of the base portion through a firstconnecting portion and the first arm portion of the second mounting armis connected to the second base portion of the base portion through asecond connecting portion, each of the first and second mounting armsextending in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms; wherein the case hasfirst and second mounting portions; wherein the second arm portion ofthe first mounting arm is mounted on the first mounting portion of thecase and the second arm portion of the second mounting arm is mounted onthe second mounting portion of the case; and wherein the lid isconnected to the case to cover an open end of the case.
 124. The quartzcrystal oscillator according to claim 120; wherein an overall length ofthe quartz crystal resonator is less than 2.1 mm; wherein the baseportion has the length less than 0.5 mm; wherein each of the first andsecond tuning fork arms comprises a first vibrational portion having afirst width and a first length, and a second vibrational portion havinga second width greater than the first width and a second length greaterthan the second width and less than the first length, the firstvibrational portion of each of the first and second tuning fork armshaving the opposite main surfaces including a first main surface and asecond main surface; wherein the at least one groove comprises a grooveformed in at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms, a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ and a second distance in the width direction of the groovemeasured from a second outer edge opposite the first outer edge of thegroove to a second outer edge opposite the first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₃; wherein a width of the groove formed in the at least one of thefirst and second main surfaces of the first vibrational portion of eachof the first and second tuning fork arms is greater than each of thefirst distance W₁ and the second distance W₃ and less than 0.07 mm;wherein each of the first distance W₁ and the second distance W₃ is lessthan 0.015 mm; wherein the base portion has a first base portion, asecond base portion and two cut portions formed between the first andsecond base portions, each of the first and second tuning fork armsbeing connected to the first base portion of the base portion; wherein afirst mounting arm is connected to the second base portion of the baseportion through a first connecting portion and a second mounting arm isconnected to the second base portion of the base portion through asecond connecting portion so that each of the first and second mountingarms extends in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms; wherein the case hasfirst and second mounting portions; wherein the first mounting arm ismounted on the first mounting portion of the case and the secondmounting arm is mounted on the second mounting portion of the case; andwherein the lid is connected to the case to cover an open end of thecase.
 125. The quartz crystal oscillator according to claim 124; whereinthe overall length of the quartz crystal resonator is within a range of0.8 mm to 1.95 mm and the length of the base portion is within a rangeof 0.015 mm to 0.49 mm; wherein each of the first and second mountingarms has a length within a range of 0.3 mm to 1.85 mm and a width; andwherein the first width of the first vibrational portion of each of thefirst and second tuning fork arms is greater than the width of each ofthe first and second mounting arms.
 126. The quartz crystal oscillatoraccording to claim 124; wherein the length of the base portion is withina range of 0.015 mm to 0.49 mm; wherein a first electrode is disposed ona surface of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms; wherein a second electrode isdisposed on a surface of each of the first and second connectingportions; wherein a third electrode is disposed on a surface of each ofthe first and second mounting arms; wherein a fourth electrode isdisposed on a surface of each of the first and second mounting portionsof the case; wherein the first electrode disposed on the surface of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of one of the first and second tuningfork arms is connected to the third electrode disposed on the surface ofthe first mounting arm through the second electrode disposed on thesurface of the first connecting portion; wherein the third electrodedisposed on the surface of the first mounting arm is connected to thefourth electrode disposed on the surface of the first mounting portionof the case; wherein the first electrode disposed on the surface of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of the other of the first and secondtuning fork arms is connected to the third electrode disposed on thesurface of the second mounting arm through the second electrode disposedon the surface of the second connecting portion; wherein the thirdelectrode disposed on the surface of the second mounting arm isconnected to the fourth electrode disposed on the surface of the secondmounting portion of the case; and wherein the fourth electrode disposedon the surface of the first mounting portion of the case has anelectrical polarity opposite to an electrical polarity of the fourthelectrode disposed on the surface of the second mounting portion of thecase.
 127. A quartz crystal oscillator according to claim 118; whereinthe opposite main surfaces of each of the first and second tuning forkarms have a first main surface and a second main surface; wherein the atleast one groove comprises a groove formed in at least one of the firstand second main surfaces of each of the first and second tuning forkarms, a first distance in the width direction of the groove measuredfrom a first outer edge of the groove to a first outer edge of thecorresponding one of the first and second tuning fork arms being definedby W₁ , a second distance in the width direction of the groove measuredfrom a second outer edge opposite the first outer edge of the groove toa second outer edge opposite the first outer edge of the correspondingone of the first and second tuning fork arms being defined by W₃;wherein a width of the groove formed in the at least one of the firstand second main surfaces of each of the first and second tuning forkarms is greater than each of the first distance Wi and the seconddistance W3; wherein a first electrode is disposed on a surface of thegroove formed in the at least one of the first and second main surfacesof each of the first and second tuning fork arms and a second electrodeis disposed on a side surface of each of the first and second tuningfork arms; wherein the first electrode disposed on _(t)he surface of thegroove formed in the at least one of the first and second main surfacesof the first tuning fork arm is connected to the second electrodedisposed on the side surface of the second tuning fork arm, and thefirst electrode disposed on the surface of the groove formed in the atleast one of the first and second main surfaces of the second tuningfork arm is connected to the second electrode disposed on the sidesurface of the first tuning fork arm so that the first electrodedisposed on the surface of the groove formed in the at least one of thefirst and second main surfaces of the first tuning fork arm has anelectrical polarity opposite to an electrical polarity of the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the second tuning fork arm,and so that the second merit value M₂ is less than 30; wherein the casehas a mounting portion; wherein the base portion is mounted on themounting portion of the case; and wherein the lid is connected to thecase to cover an open end of the case.
 128. The quartz crystaloscillator according to claim 120; wherein each of the first and secondtuning fork arms comprises a first vibrational portion having a firstwidth and a first length, and a second vibrational portion having asecond width greater than the first width and a second length less thanthe first length, the first vibrational portion of each of the first andsecond tuning fork arms having the opposite main surfaces including afirst main surface and a second main surface; wherein the at least onegroove comprises a groove formed in at least one of the first and secondmain surfaces of the first vibrational portion of each of the first andsecond tuning fork arms, a first distance in the width direction of thegroove measured from a first outer edge of the groove to a first outeredge of the corresponding one of the first and second tuning fork armsbeing defined by W₁ and a second distance in the width direction of thegroove measured from a second outer edge opposite the first outer edgeof the groove to a second outer edge opposite the first outer edge ofthe corresponding one of the first and second tuning fork arms beingdefined by W₃; wherein a width of the groove formed in the at least oneof the first and second main surfaces of the first vibrational portionof each of the first and second tuning fork arms is greater than each ofthe first distance W₁ and the second distance W₃ and less than 0.07 mm;wherein each of the first distance W₁ and the second distance W₃ is lessthan 0.015 mm; wherein an overall length of the quartz crystal resonatoris less than 2.1 mm; wherein the length of the base portion is less than0.5 mm; wherein the base portion has a first base portion, a second baseportion and at least one cut portion formed between the first and secondbase portions, each of the first and second tuning fork arms beingconnected to the first base portion of the base portion, the case havinga mounting portion; wherein the second base portion of the base portionis mounted on the mounting portion of the case; and wherein the lid isconnected to the case to cover an open end of the case.
 129. The quartzcrystal oscillator according to claim 120; wherein each of the first andsecond tuning fork arms comprises a first vibrational portion having afirst width and a first length, and a second vibrational portion havinga second width greater than the first width and a second length greaterthan the second width and less than the first length, the firstvibrational portion of each of the first and second tuning fork armshaving the opposite main surfaces including a first main surface and asecond main surface; wherein the at least one groove comprises a grooveformed in at least one of the first and second main surfaces of thefirst vibrational portion of each of the first and second tuning forkarms, the first vibrational portion of each of the first and secondtuning fork arms having a first side surface and a second side surfaceopposite the first side surface; wherein a first electrode is disposedon a surface of the groove formed in the at least one of the first andsecond main surfaces of the first vibrational portion of each of thefirst and second tuning fork arms, and a second electrode is disposed oneach of the first and second side surfaces of the first vibrationalportion of each of the first and second tuning fork arms so that thefirst merit value M₁ of the fundamental mode of vibration is greaterthan the second merit value M₂ of the second overtone mode of vibration;wherein the case has a mounting portion; wherein the base portion ismounted on the mounting portion of the case; and wherein the lid isconnected to the case to cover an open end of the case.
 130. The quartzcrystal oscillator according to claim 129; wherein a length of thegroove formed in the at least one of the first and second main surfacesof the first vibrational portion of each of the first and second tuningfork arms is within a range of
 0. 45 mm to 1.25 mm; wherein the baseportion has a first base portion, a second base portion and a cutportion formed between the first and second base portions, each of thefirst and second tuning fork arms being connected to the first baseportion of the base portion; wherein the second base portion of the baseportion is mounted on the mounting portion of the case; and wherein thefirst electrode disposed on the surface of the groove formed in the atleast one of the first and second main surfaces of the first vibrationalportion of the first tuning fork arm is connected to the secondelectrode disposed on each of the first and second side surfaces of thefirst vibrational portion of the second tuning fork arm, and the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the first vibrationalportion of the second tuning fork arm is connected to the secondelectrode disposed on each of the first and second side surfaces of thefirst vibrational portion of the first tuning fork arm so that the firstelectrode disposed on the surface of the groove formed in the at leastone of the first and second main surfaces of the first vibrationalportion of the first tuning fork arm has an electrical polarity oppositeto an electrical polarity of the first electrode disposed on the surfaceof the groove formed in the at least one of the first and second mainsurfaces of the first vibrational portion of the second tuning fork arm,and so that the second merit value M₂ of the second overtone mode ofvibration of quartz crystal resonator is less than 30 and thecapacitance ratio r of the second overtone mode of vibration thereof isgreater than
 1500. 131. The quartz crystal oscillator according to claim129; wherein the base portion has a first base portion including a firstlength, a second base portion and a third base portion connecting thefirst base portion to the second base portion, two cut portions beingformed between the first and second base portions, each of the first andsecond tuning fork arms being connected to the first base portion of thebase portion; wherein the second base portion of the base portion ismounted on the mounting portion of the case; and wherein the secondlength of the second vibrational portion of each of the first and secondtuning fork arms is greater than the first length of the first baseportion of the base portion.
 132. The quartz crystal oscillatoraccording to claim 129; wherein each of the first and second tuning forkarms has a length less than 1.56 mm; wherein the first width of thefirst vibrational portion of each of the first and second tuning forkarms is less than 0.1 mm; wherein the base portion has a first baseportion, a second base portion and two cut portions formed between thefirst and second base portions, each of the first and second tuning forkarms being connected to the first base portion of the base portion;wherein two mounting arms having first and second mounting arms areconnected to the second base portion of the base portion; wherein thefirst mounting arm is connected to the second base portion of the baseportion through a first connecting portion and the second mounting armis connected to the second base portion of the base portion through asecond connecting portion. each of the first and second mounting armsextending in a common direction with the first and second tuning forkarms outside the first and second tuning fork arms. and having a widthand a length within a range of 0.3 mm to 1.85 mm; and wherein the widthof each of the first and second mounting arms is different from thefirst width of the first vibrational portion of each of the first andsecond tuning fork arms.
 133. The quartz crystal oscillator according toclaim 129; wherein an overall length of the quartz crystal resonator isless than 2.1 mm; wherein the base portion has the length less than 0.5mm; wherein the base portion has a first base portion including a firstlength and a second base portion, a cut portion being formed between thefirst and second base portions, each of the first and second tuning forkarms being connected to the first base portion of the base portion;wherein the second base portion of the base portion is mounted on themounting portion of the case; and wherein the second length of thesecond vibrational portion of each of the first and second tuning forkarms is greater than the first length of the first base portion of thebase portion.